System and method for a hybrid clock and proxy auction

ABSTRACT

The present invention primarily concerns hybrid auctions that may, for example, combine a clock auction with a proxy auction. Hybrid auctions include multi-item auctions that comprise at least two phases of package auctions: an earlier phase in which bidders participate in a clock auction (or other dynamic auction); and a later phase in which bidders participate in a proxy auction (or some other package auction). By combining the earlier phase and the later phase as in some of the embodiments described herein, it is possible to combine the advantages of the dynamic auction and the advantages of the sealed-bid package auction. In particular, if the earlier phase is a clock auction and the later phase is a proxy auction, then the resulting hybrid auction will combine the transparency and simplicity of the clock auction with the efficient outcome and competitive revenues of the proxy auction.

RELATED APPLICATIONS

This application claims the benefit of the filing dates of the followingco-pending patent applications:

Ausubel, Cramnton and Milgrom, “SYSTEM AND METHOD FOR A HYBRID CLOCK ANDPROXY AUCTION”, U.S. Provisional Patent Application, Ser. No. 60/517,380filed Nov. 6, 2003.

Ausubel and Milgrom, “SYSTEM AND METHOD FOR A DYNAMIC AUCTION WITHPACKAGE BIDDING”, U.S. patent application, Ser. No. 10/432,250 filed May22, 2003, which is the U.S. National Stage Entry of InternationalApplication Number: PCT/US01/43838, filed Nov. 23, 2001.

Ausubel, Cramton and Jones, “SYSTEM AND METHOD FOR AN AUCTION OFMULTIPLE TYPES OF ITEMS”, U.S. patent application, Ser. No. 10/467,868filed Aug. 13, 2003, which is the U.S. National Stage Entry ofInternational Application Number: PCT/US02/16937, filed May 31, 2002.

FIELD OF THE INVENTION

The present invention relates to improving computer-implemented auctionsand, more particularly, to computer implementation of a hybrid auctioncombining elements of a clock auction and a proxy auction.

BACKGROUND OF THE INVENTION

There are many situations where items need to be allocated among two ormore parties. Often, the items are related, so that a party's value fora package of items is different from the sum of the party's values forthe separate items. Auctions (or auction-like processes) are often usedto allocate items among two or more parties, but the design of anefficient auction is a technically difficult problem when the items arerelated.

When items are related, they are often related so as to be complementsor substitutes. Two items are said to be complements when they are usedtogether; more precisely, items A and B are complements when reducingthe price of item A causes the demand for item B to increase. Two itemsare said to be substitutes when one is used in place of the other; moreprecisely, items A and B are substitutes when reducing the price of itemA causes the demand for item B to fall.

One example of allocating complementary items may be the case ofallocating licenses for telecommunications spectrum. A government mayoffer a variety of spectrum licenses, each covering a specifiedbandwidth in a specified geographic area. A wireless telephone companyseeking spectrum rights may find that it requires a minimum of 20 MHz ofbandwidth in a given geographic area, in order to be able to provide auseful service. Thus, two 10 MHz licenses covering the same geographicarea may be complements. At the same time, a wireless telephone companymay realize synergies from serving two geographically adjacent markets,so licenses for two adjacent markets may also be complements.

One example of allocating substitute items may be the case of allocatingfinancial instruments. A government may offer three-month, six-month,one-year, two-year and five-year debt securities. For a buyer ofgovernment securities, these different durations are likely to besubstitutes to at least a certain extent, since they each provide a safeinvestment opportunity for the next three months. For the government,too, these different durations are likely to be substitutes to at leasta certain extent, since they each provide a vehicle for financing thegovernment's debt for the next three months.

In some of the most difficult situations some items are complements andother items are substitutes within the same allocation problem. Forexample, consider the case of allocating capacity at one or morecapacity-onstrained airport. The airport authority may offer a varietyof landing slots and takeoff slots at different times of the day.Various slots may be complementary goods: for example, an airline islikely to desire a takeoff slot about one hour after each landing slot,and an airline may desire several takeoff and landing slots bunchedtogether in order to be able to maintain a “hub-and-spoke” operation.Also, various slots may be substitute goods: for example, an airline maybe able to use a 10:00 am landing slot in place of a 9:00 am landingslot, although the former slot may not be anywhere as valuable as thelatter slot. The slots at two airports serving the same city are likelyto be substitutes for one another, while the slots at two ends of aheavily trafficked route are likely to be complementary goods.

In the last few years, there has been growing interest in auctionprocesses that allow bidders much greater freedom to name the packageson which they bid during the auction. Such auctions, which may determinethe packaging, pricing and allocation decisions, can be called “packageauctions,” “combinatorial auctions,” or “auctions with package bidding.”Typically, bidders in these auctions describe the packages that theywish to acquire and make bids for the named packages.

One especially promising version of a package auction is a clockauction. It is an iterative auction process in which the auctioneerannounces prices, bidders respond with quantities, the prices areadjusted according to the relation between the quantities bid and thequantities being auctioned, and the process is allowed to repeat. Suchauction processes are particularly effective in allocating multipleunits of multiple types of goods. (For a longer discussion, see “Systemand Method for an Auction of Multiple Types of Items,” InternationalPatent Application No. US02/16937.)

A second especially promising version of a package auction is aproxyauction. It is effectively a sealed-bid auction. The bidder inputsvaluation information into a proxy agent, so that the proxy agentpossesses information concerning the valuation of some or all of thepossible packages. The proxy agent then submits package bids on behalfof the bidder, selecting one or more packages that optimize thedifference between the bidder's value and the amount that can be bid forthe package. The auctioneer then selects provisionally-winning bids bysolving the optimization problem of selecting bids, at most one fromeach bidder, that optimize revenues subject to a feasibility constraint.Proxy agents for bidders who are not selected as provisional winnersthen submit new bids, and the process continues until the bidders whoare not provisional winners have no profitable bids remaining to beplaced. (For a longer discussion, see “System and Method for a DynamicAuction with Package Bidding,” International Patent Application No.US01/43838.)

SUMMARY OF THE INVENTION

The present invention primarily concerns hybrid auctions that may, forexample, combine a clock auction with a proxy auction. Hybrid auctionsinclude multi-item auctions that comprise at least two phases of packageauctions: an earlier phase in which bidders participate in a clockauction (or other dynamic auction); and a later phase in which biddersparticipate in a proxy auction (or some other package auction). Bycombining the earlier phase and the later phase as in some of theembodiments described herein, it is possible to combine the advantagesof the dynamic auction and the advantages of the sealed-bid packageauction. In particular, if the earlier phase is a clock auction and thelater phase is a proxy auction, then the resulting hybrid auction willcombine the transparency and simplicity of the clock auction with theefficient outcome and competitive revenues of the proxy auction.

In a first preferred embodiment, the present invention is acomputer-implemented method for a hybrid auction that includes anearlier phase, in which bidders have two or more opportunities to submitpackage bids where the price associated with a given package isdetermined by linear pricing, and a later phase, in which bidders submitpackage bids and the outcome is determined to be a bidder-optimal coreoutcome (relative to the submitted package bids).

In a second preferred embodiment, the present invention is a computersystem for conducting an auction according to the method of the firstpreferred embodiment. The system includes means for receiving bids,means for determining an outcome, and means for outputting.

In a third preferred embodiment, the present invention is:

-   A computer implemented method for conducting an auction of a    plurality of items wherein at least one computer receives bids and    determines an allocation of at least one of the items, the auction    including a dynamic auction phase followed by a later phase, the    later phase comprising a package auction, the method comprising:    -   a) implementing the dynamic auction phase on a computer, said        dynamic auction phase comprising:        -   a1) receiving bids from at least one bidder, said bids            including at least an indicator of at least one of the            items;        -   a2) determining whether the dynamic auction phase of the            auction should continue, based on received bids;        -   a3) outputting auction information; and        -   a4) repeating a1)-a3) if the dynamic auction phase of the            auction is determined to continue;    -   b) changing from the dynamic auction phase to the later phase,        following a determination not to continue the dynamic auction        phase; and    -   c) implementing the later phase of the auction on a computer,        the later phase comprising a package auction, said later phase        comprising:        -   c1) receiving bids from at least one bidder, said bids            including at least an indicator of a package of items and an            associated price for the package; and        -   c2) determining an allocation of at least one of the items            to one of the bidders based on received bids.

In a fourth preferred embodiment, the present invention is:

-   A computer implemented system for conducting an auction of a    plurality of items wherein at least one computer receives bids and    determines an allocation of at least one of the items, the auction    including a dynamic auction phase followed by a later phase, the    later phase comprising a package auction, the system comprising:    -   a) means for implementing the dynamic auction phase on a        computer, said means for implementing the dynamic auction phase        comprising:        -   a1) means for receiving bids from at least one bidder, said            bids including at least an indicator of at least one of the            items;        -   a2) means for determining whether the dynamic auction phase            of the auction should continue, based on received bids;        -   a3) means for outputting auction information; and        -   a4) means for repeating a1)-a3) if the dynamic auction phase            of the auction is determined to continue;    -   b) means for changing from the dynamic auction phase to the        later phase, following a determination not to continue the        dynamic auction phase; and    -   c) means for implementing the later phase of the auction on a        computer, the later phase comprising a package auction, said        means for implementing said later phase comprising:        -   c1) means for receiving bids from at least one bidder, said            bids including at least an indicator of a package of items            and an associated price for the package; and        -   c2) means for determining an allocation of at least one of            the items to one of the bidders based on received bids.

Various systems and methods in the art facilitate the operation ofcomputer-implemented auctions. The implementation of auctions oncomputers holds numerous advantages over the earlier art. It facilitatesthe simultaneous auctioning—in a single, combined auction process—of aplurality of items that are related, for example, in the sense thatbidders may value the items as substitutes or complements. It permits adynamic bidding process for such a plurality of items, in which biddersin diverse locations across the continent or the globe are able toactively participate and to receive feedback in real time about theiropponents' bids. It enables the practical introduction of clockauctions, proxy auctions, and other forms of package bidding. And inaccomplishing the above, it encourages bidders to bid aggressively andstraightforwardly for the packages they want, incorporating allavailable information, and resulting in items being allocated to thebidders who value them the most, while also ensuring a competitive pricefor the seller or sellers.

The present invention is useful for “reverse auctions” or “procurementauctions”conducted by or for buyers to acquire various kinds of items orresources, “standard auctions” conducted by or for sellers in whichitems are offered for sale, and “exchanges” in which both buyers andsellers place bids. Although terms such as “items or quantitiesdemanded” (by a bidder) and “demand curve” (of a bidder) are used todescribe the present invention, the terms “items or quantities offered”(by a bidder) and “supply curve” (of a bidder) are equally applicable.In some cases, this is made explicit by the use of both terms, or by theuse of the terms “items or quantities transacted” (by a bidder) and“transaction curve” (of a bidder). The term “items or quantitiestransacted” includes both “items or quantities demanded” and “items orquantities offered”. The term “bid” includes both offers to sell andoffers to buy. The term “transaction curve” includes both “demand curve”and “supply curve”. Moreover, any references to “items or quantitiesbeing offered” includes both “items or quantities being sold” by theauctioneer, in the case this is a standard auction for selling items, aswell as “items or quantities being bought or procured” by theauctioneer, in the case this is a procurement auction or reverse auctionfor buying or acquiring items.

Moreover, while standard auctions to sell typically involve ascendingprices, the present invention may utilize descending prices, and moregenerally, prices that ascend and/or descend.

Throughout this document, the terms “objects”, “items”, “units” and“goods” are used essentially interchangeably. The inventive system andmethod may be used both for tangible objects, such as real or personalproperty, and intangible items, such as telecommunications licenses orelectric power. The inventive system and method may be used in auctionswhere the auctioneer is a seller, buyer or broker, the bidders arebuyers, sellers or brokers, and for auction-like activities which cannotbe interpreted as selling or buying. The inventive system and method maybe used for items including, but not restricted to, the following:public-sector bonds, bills, notes, stocks, and other securities orderivatives; private-sector bonds, bills, notes, stocks, and othersecurities or derivatives; communication licenses and spectrum rights;clearing, relocation or other rights concerning encumbrances of spectrumlicenses; electric power and other commodity items; rights for terminal,entry, exit or transmission capacities or other rights in gas pipelinesystems; airport landing and takeoff rights or other transportationrights; emission allowances and pollution permits; and other goods,services, objects, items or other property, tangible or intangible. Itmay also be used for option contracts on any of the above. It may beused in initial public offerings, secondary offerings, and in secondaryor resale markets.

The network used, if any, can be any system capable of providing thenecessary communication to/from a Bidding Information Processor (BIP), aBidding Terminal (BT), and an Auctioneer's Terminal (AT). The networkmay be a local or wide area network such as, for example, Ethernet,token ring, the Internet, the World Wide Web, the informationsuperhighway, an intranet or a virtual private network, or alternativelya telephone system, either private or public, a facsimile system, anelectronic mail system, or a wireless communications system, orcombinations of the foregoing.

The following patents are related to the present invention:

-   Ausubel, Lawrence M., U.S. Pat. No. 5,905,975, May 1999.-   Ausubel, Lawrence M., U.S. Pat. No. 6,021,398, February 2000.-   Ausubel, Lawrence M., U.S. Pat. No. 6,026,383, February 2000.-   McAfee, R. Preston and Paul Milgrom, U.S. Pat. No. 6,718,312, April    2004.

The following other references may be related to the present invention:

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BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graphical depiction of the architecture of an exemplarycomputer system in accordance with an embodiment of the invention;

FIG. 2 is a graphical depiction of another exemplary computer system inaccordance with an embodiment of the invention;

FIG. 3 is a detail of one element of the computer system of FIG. 2;

FIG. 4 is a flow diagram of an exemplary hybrid auction in accordancewith an embodiment of the invention;

FIG. 5 is a flow diagram in greater detail of an exemplary hybridauction in accordance with an embodiment of the invention;

FIGS. 6 a and 6 b are flow diagrams illustrating, in greater detail,elements of the flow diagram of FIG. 5;

FIGS. 7 a, 7 b and 7 c are flow diagrams illustrating, in greaterdetail, elements of the flow diagram of FIG. 5;

FIGS. 8 a and 8 b are flow diagrams illustrating, in greater detail,elements of the flow diagram of FIG. 5;

FIGS. 9 a, 9 b and 9 c are flow diagrams illustrating, in greaterdetail, elements of the flow diagram of FIG. 5;

FIG. 10 is a flow diagram illustrating, in greater detail, an element ofthe flow diagram of FIG. 5;

FIG. 11 is a flow diagram illustrating, in greater detail, an element ofthe flow diagram of FIG. 5;

FIG. 12 is a graphical depiction of the architecture of an exemplaryauction system in which bidding is intermediated by proxy agents, inaccordance with an embodiment of the invention;

FIG. 13 is a flow diagram of an exemplary proxy auction phase, inaccordance with an embodiment of the invention;

FIGS. 14 a and 14 b are flow diagrams illustrating, in greater detail,elements of the flow diagram of FIG. 13;

FIGS. 15 a and 15 b are flow diagrams illustrating, in greater detail,elements of the flow diagram of FIG. 13; and

FIG. 16 is a flow diagram illustrating, in greater detail, an element ofthe flow diagram of FIG. 5.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Overall Structure of Auction System

Earlier auction methods and systems are described in U.S. Pat. Nos.5,905,975, 6,021,398 and 6,026,383. The following description willdetail the flow of the novel features of the preferred embodiments ofthe present method and system for a hybrid auction combining a clockauction (or other dynamic auction) with a proxy auction (or otherauction allowing package bidding).

Before describing the auction process in detail, reference is made toFIG. 1 to describe the architecture of an exemplary computer system inaccordance with an embodiment of the present invention. In the graphicaldepiction of FIG. 1, the computer system consists of multiple bidder andauctioneer computers or terminals 20 a-n and 30 communicating with theserver (or auction computer) 10 over a network 40. The computers orterminals 20 a-n are employed by bidders, the computer or terminal 30 isemployed by the auctioneer, and the server 10 is the auction computer.The server 10 consists of a CPU 11, memory 12, a data storage device 13,a communications interface 14, a clock 15, an operating system 16, andan auction program 17. In one embodiment, the system architecture is asa client-server system: the auction computer is a server; and the bidderand auctioneer computers are clients.

FIG. 2 is another graphical depiction of an exemplary computer system inaccordance with an embodiment of the present invention. The auctionsystem of FIG. 2 includes an auction computer 60 (sometimes alsoreferred to as a Bidding Information Processor or BIP), a plurality ofuser systems 70 a, 70 b and so on (sometimes also referred to as BidderTerminal or BT), each user system 70 a-n representing an individualbidder, and a user system 80 (sometimes also 10 referred to as anAuctioneer Terminal or AT). The systems 60, 70 a-n, and 80 communicateover a network 90. The network represents any system capable ofproviding the necessary communication to/from BIP, BT, and AT. Thenetwork may be a local or wide area network such as, for example,Ethernet, token ring, the Internet, the World Wide Web, the informationsuperhighway, an intranet or a virtual private network, or alternativelya telephone system, either private or public, a facsimile system, anelectronic mail system, or a wireless communications system. Each of thesystems 60, 70 a-n, and 80 may include a typical user interface 65, 75a-n, 85 for input/output which may include a conventional keyboard,display, and other input/output devices. Within each of the systems, theuser interface (65, 75 a-n, 85) is coupled to a network interface (64,74 a-n, 84), which in turn communicates via the network 90. Both theuser interface 20 and network interface connect, at each system, to aCPU (62, 72 a-n, 82). Each system includes a memory (66, 76 a-n, 86).The BIP 60 also includes a clock 61 and a data storage device 63, whichwill ordinarily contain a database. (However, in some embodiments thedatabase might instead be stored in memory 66.) The memory 66 of the BIP60 can further be broken down into a program 67, data 68 and anoperating system 69. The memory (76 a-n, 86) of the BT's 70 a-n and theAT 80 may include a web browser (for example, Internet Explorer orNetscape) (79, 89) or other general-purpose software, but notnecessarily any computer program specific to the auction process. Ineach system the CPU (62, 72 a-n, 82) represents a source of intelligencewhen executing instructions from the memory (66, 76 a-n, 86) so thatappropriate input/output operations via the user interface and thenetwork interface take place as is conventional in the art. Theparticular steps used in implementing the inventive auction system andmethod are described in more detail below. In one embodiment, each ofthe user systems is a personal computer or workstation.

FIG. 3 is a more detailed illustration of an exemplary BIP 60 showingdetails of the database. As discussed for FIG. 2, the database isordinarily stored on a data storage device 63, although in someembodiments it might instead be stored in memory 66. As depicted in FIG.3, the database includes provision for creating, storing, and retrievingrecords representing Items in the Auction 63-1, Status of the Items inthe Auction 63-2, Auction Timetable 63-3, Current Price(s) 634, List ofBidder ID's 63-5, List of Passwords 63-6, Bidding History 63-7, andConstraints on Bids 63-8. The particular set of data required for anyparticular auction and the format of that datum or data (such as scalar,vector, list, etc.) is more particularly specified by the detaileddescription of that auction.

Bidders, Items, Types and Groups

There are n bidders (n≧1), often superscripted or subscripted by i (i=1,. . . , n), participating in the auction. Let Ω denote the set of itemsthat are included in the auction. The aim of the auction is to allocate,among the bidders, the elements of the set Ω.

Some of the most useful embodiments of the present invention apply insituations where some of the elements of the set Ω are identical orclose substitutes to one another. A type of item comprises a (nonempty)subset of Ω, such that any two items within the same type are identicalitems or close substitutes. Meanwhile, types are defined so that any twoitems of different types exhibit significant differences in time,location or any other product characteristics. There may be a singleunit or multiple units within a given type. Whenever we state that thereare “unique goods,” we are referring to the case where there is only asingle unit within every type.

In what follows, we will assume that there are m types of items (m≧1)included in the auction. Whenever we state that there is a “plurality oftypes of items,” we are referring to the case where m≧2. There are oftensaid to be “heterogeneous” goods or “dissimilar items” in the case wherem≧2. Whenever we state that there are “homogeneous” goods or “identicalobjects,” we are referring to the case where m=1. Whenever we state thatthere is a “plurality of bidders,” we are referring to the case wherethe number, n, of bidders participating in the auction satisfies n≧2.

Some of the useful embodiments of the present invention apply insituations where two or more of the types of items can be usefullygrouped together. A group G of types comprises a (nonempty) subset of{1, . . . , m}, the set of all types of items. Typically, the reasonthat two types of items are included in the same group is that they arerelated, for example, they may be contracts for provision of the samecommodity, covering different but overlapping time periods.

If types of items are classified into groups, then our notation will bethat there are h (h≧1) groups. There is no requirement that each groupcontains the same number of types of items, but this often happens to bethe case. Whenever we need notation indicating the number of types ofitems contained in the exemplary group G, our notation will be thatthere are g (g≧1) types of items in group G.

Some examples of “types” and “groups,” in situations where there may besignificant commercial possibilities for embodiments of the presentinvention, include the following:

-   -   Treasury bills or other securities: A government or central bank        may wish to auction 3-month, 6-month and 12-month Treasury        securities (for example, with the same starting date) together.        Thus, there is one group of types of items (h=1), and it        contains three types of items (g=3). In total, there are three        types of items (m=3).    -   Electricity contracts: An electric generating company may wish        to simultaneously auction some forward contracts or options        contracts for base-load and peak-load electricity generation,        with durations of 2 months, 3 months, 6 months, 12 months, 24        months and 36 months, respectively. Thus, there are two groups        of types of items (h=2), each containing six types of items        (g=6). In total there are 2×6 =12 types of items (m=12).    -   Two unrelated, heterogeneous consumer commodities (e.g., apples        and oranges). There are two groups of types of items (h=2), each        containing just one type of item (g=1), for a total of two types        of items (m=2).        Packages and Package Bids

In many preferred embodiments of the present invention, bids comprisepairs, (S, P), where S⊂Ω is a package, i.e., a subset of the set of allitems being auctioned, and P is a price at which the bidder is offeringto transact for the entire package S. Stated differently, a bidcomprises a package of items and an associated price for the entirepackage. Such a bid comprising a pair, (S, P), is defined to be apackage bid.

In the event that the elements of the set Ω have been classified intotypes, a package can be identified by a quantity vector (Q₁, . . . ,Q_(m)) of each of the m (m≧1) types of items. A package bid for bidder iwould then comprise a quantity vector (Q₁ ^(i), . . . , Q_(m) ^(i)) anda price, P, at which bidder i is offering to transact for the entire setidentified by the quantity vector. Such a bid (Q₁ ^(i), . . . , Q_(m)^(i); P) is defined to be a package bid in the “type” notation.

Clock Auctions

A clock auction is an auction process that proceeds as follows. First, aprice vector indicating the price of each type of item is transmitted tobidders (i.e., bidder terminals). Second, a bidder responds with aquantity vector indicating the quantity of each respective type of itemthat the bidder wishes to transact at the current price vector. Acomputer determines whether the auction shoudl continue or terminate,and if the decision is to continue, the price vector is adjusted and theprocess is repeated.

More precisely, consider a situation where the items included in theauction have been classified into types. The current price vectorcomprises a vector, (P₁, . . . , P_(m)), whose components represent theprices (per unit) for the m respective types of items. Each bidder iselects one or more package bids by selecting a quantity vector, (Q₁^(i), . . . , Q_(m) ^(i)), whose components indicate the quantities thatbidder i is willing to buy (in the case of an auction to sell) or tosell (in the case of an auction to buy) at the current price vector forthe m respective types of items. The quantity vector thus identifies thepackage of items that bidder i desires to transact. The associatedprice, P^(i), at which bidder i is offering to buy or sell the packageis implied by:

$P^{i} = {\overset{m}{\sum\limits_{k = 1}}{p_{k}{Q_{k}^{i}.}}}$

A bid of bidder i in a clock auction comprises a quantity vector, (Q₁^(i), . . . , Q_(m) ^(i)), together with the implied price, P^(i),calculated at the current price vector. Depending on the particularembodiment of the present invention, a bidder may be permitted to selecta single bid at each current price vector, or a bidder may be permittedto select more than one bid at each current price vector.

The bidding history comprises the current price vector and the bidsassociated with the current time and all earlier times in the currentauction.

The available quantities may, in principle, be specified for each typeof item or for each group of types of items. In the text that follows,we will usually specify the available quantity for each type of item.The available quantity for type k of item will be denoted Q _(k), andthis refers, in the case of an auction to sell, to the overall quantityof items of type k to be offered for sale in the auction or, in the caseof an auction to buy (i.e., a procurement auction or a “reverseauction”), to the overall quantity of items of type k to be bought inthe auction. The vector of available quantities for all types will bedenoted ( Q ₁, . . . , Q _(m)). (If, instead, we wish to indicate theavailable quantity for group G of types of items, this will be denotedby Q ^(G), and the vector of available quantities for all groups will bedenoted ( Q ¹, . . . , Q ^(h)).) Optionally, the available quantitiesmay be allowed to depend on the prices, or otherwise be contingent onthe progress of the auction.

With this terminology defined, a clock auction is a dynamic auctionprocedure whereby: the current price vector is announced to bidders; thebidders each respond with quantity vectors indicating their bids; theauctioneer determines whether the auction should continue based on thebidding history and the available quantities; the auctioneer updates thecurrent price vector based on the bidding history and the availablequantities, and the process repeats, if it is determined that theauction should continue; and the auctioneer allocates the items amongthe bidders and assesses payments among the bidders based on the biddinghistory and the available quantities, if it is determined that theauction should not continue.

Observe that a “clock auction” differs from a standard ascending-bidelectronic auction in the following important sense. In standardascending-bid electronic auctions—such as in the Federal CommunicationsCommission auctions for radio communications spectrum or in eBayauctions—the bidders name prices (and, perhaps also, quantities) thatthey propose to pay for the items being auctioned, in an iterativeprocess. In a clock auction, the auctioneer sets the pace for priceincreases, and bidders respond only with quantity vectors—the associatedpayments being implied by the current price vector.

Flexible Bid Information and Proxy Agents

Flexible bid information is data that a bidder selects for present orfuture use by entering into a computer (e.g., a bidder computer or aBT), but at least some of such data is stored in a database rather thanbeing directly and immediately submitted as a bid in an auction.Flexible bid information can include a scalar value, a vector value, ora function. The flexible bid information may be an expression of which(or how many units of) item(s) a bidder is willing to purchase at agiven price(s), how much money a bidder is willing to pay for thepurchase of a given item(s), or any other expression of the value whicha bidder places on item(s) or a stopping price that a bidder places onitem(s). It may also include an expression of how much money or otherconsideration a bidder is willing to spend in aggregate for all of theitems purchased. Optionally, flexible bid information may include abidding rule that contains a limitation (e.g., “I desire up to aquantity of x at a price P, but I do not want any positive quantity atall unless I receive a minimum quantity of y”). Thus, flexible bidinformation may include one or more bidding rules that may compriseunconditional bids or contingent bids, and may include one or morefunctions from available information to bid quantities (e.g. a functionof the previous bid(s) submitted).

Within the specific context of an auction with package bidding, flexiblebid information may include valuation information, budget information,and other information. Valuation information comprises data relating oneor more subsets of the set of all items to indices of price or value,often measured in dollars or other monetary units. For example,valuation information in a package auction for the items {A,B,C} mayinclude a measure of the valuation or cost that a bidder attaches toeach of the subsets Ø, {A}, {B}, {C}, {A,B}, {A,C}, {B,C} and {A,B,C}.Alternatively, this may include a stopping price at which the bidderwishes to stop bidding for each of the respective subsets. Budgetinformation comprises data relating to an aggregate index of price orpayment, often measured in dollars or other monetary units. For example,budget information in a package auction for the items {A,B,C} mayinclude a measure of the overall budget limit or parameter for whateveritems that a given bidder may sell or buy. Other information comprisesdata relating to the auction that is neither valuation information norbudget information.

The state of the auction system refers to the full history of bids andmessages submitted by or on behalf of bidders in the auction process,the full history of messages submitted on behalf of the auctioneer, thefull history of constraints imposed by the auction system, an indicatorof which phase of the auction the process currently is in, and any otherrelevant information- about the progress of the auction. In some of thepreferred embodiments of the inventive system and method, bidders arepermitted to change or are not permitted to change their flexible bidinformation, according to rules based on the state of the auctionsystem. In that event, the state of auction system may itself include alist of the past time or times at which bidders were allowed to changetheir flexible bid information, as well as information about theprogress of the auction since this time or these times. In someembodiments the state of the auction system is limited to informationreaching the auction computer. However, in other embodiments it includesinputs from the bidder representing flexible bid information. The “stateof the auction system” is sometimes referred to, more compactly, as the“auction state information.”

The current auction information refers to the portion of the state ofthe auction system that is made available to bidders. In some preferredembodiments, the auction is conducted in discrete rounds, and biddersare provided with full information about previous rounds, so that thecurrent auction information in a given auction round may include thehistory of bids and messages submitted by or on behalf of bidders in theauction process, up until and including the previous auction round. Inother preferred embodiments, the auction is conducted in discreterounds, but bidders are provided with less than full information aboutprevious rounds, and so the current auction information in a givenauction round may include only a very abbreviated summary of the historyof bids and messages submitted by or on behalf of bidders in the auctionprocess, up until and including the previous auction round. In otherpreferred embodiments, the auction is conducted in continuous time, andthe current auction information at a given time may include the historyof bids and messages submitted by or on behalf of bidders in the auctionprocess, with some amount of time lag.

A proxy agent is a computer-implemented system which may submit bids orsend messages on behalf of a bidder, based on flexible bid information,current auction information, and/or the state of the auction system.Thus, the inputs of the proxy agent may include flexible bidinformation; and the outputs of the proxy agent may include bids ormessages. Another way to describe this is that a proxy agent may takeflexible bid information as instructions and may submit bids or sendmessages on behalf of a bidder. A proxy agent may be a subsystem of alarger computer-implemented auction system, or it may be a stand-alone,computer-implemented system that is capable of interacting with acomputer-implemented auction system.

In some embodiments of the inventive system and method, the bidding maybe intermediated by proxy agents. More precisely, a bidder may enterflexible bid information at a bidder computer or a BT, and a proxy agentmay submit bids on behalf of the bidder: this process will often bereferred to as proxy bidding. In such embodiments of the inventivesystem and method, proxy bidding may either be voluntary or mandatory.One purpose of voluntary proxy bidding is to facilitate participation bybidders in a dynamic auction. With voluntary proxy bidding, a bidder whoexpects to be busy during part or all of a dynamic auction can instructa proxy agent to bid in his (or her) place. One purpose of mandatoryproxy bidding is to limit the possibilities for collusion among bidders.For example, it may be believed that bidders can tacitly collude bymaking use of retaliatory strategies: if bidder ABC raises the high bidon an item of interest to bidder XYZ, an example of a retaliatorystrategy would be for bidder XYZ to respond by raising the high bid onan item of interest to bidder ABC. With mandatory proxy bidding, theauctioneer may require bidder XYZ to input his (or her) valuationinformation into a proxy agent that is incapable of carrying out aretaliatory strategy, effectively limiting the possibilities forcollusion among bidders.

Furthermore, in an auction system or method with proxy bidding, a biddermay be allowed to make changes to the flexible bid information that isused by its proxy agent, or a bidder may not be allowed to make suchchanges. Obviously, a restriction on changes to the flexible bidinformation has the greatest force in an auction system where proxybidding is mandatory. Moreover, the setting on an auction system as towhether a bidder is allowed to make changes may itself be changed overtime (or status), and may depend on the history of bidding (or on theidentity) of the bidder. For example, bidder i may be allowed to changeits flexible bid information early in the auction, but the same bidder imay not be allowed to make changes in its flexible bid informationbeyond a certain time in the auction. The change in setting for bidder imay depend on the course of bidder i's bidding in the auction. Forexample, the setting that bidder i is not allowed to make furtherchanges to its flexible bid information may be triggered by the factthat bidder i (or its proxy agent) has submitted insufficiently few newbids between time t and time t+1 of the auction.

Proxy Auctions

A proxy auction is an auction process with proxy bidding. Observe thatan auction process with mandatory proxy bidding—and in which a bidder isnot allowed to make changes to the flexible bid information that is usedby its proxy agent—is observationally equivalent to a sealed-bidauction. In addition, it will be described below that the outcome of aproxy auction will always be approximately a core outcome with respectto the valuation information submitted by bidders. Thus, in the textthat follows, the term “proxy auction” will also be used to include anysealed-bid package auction that results in approximately a core outcomewith respect to the bids submitted by bidders.

Hybrid Auctions

A hybrid auction is an auction for at least two items that includes twophases of auctions: an earlier phase in which bidders participate in adynamic auction; and a later phase in which bidders participate in apackage auction. One exemplary hybrid auction (which is a preferredembodiment of the present invention) comprises an earlier phase in whichbidders participate in a clock auction and a later phase in whichbidders participate in a proxy auction.

FIG. 4 is a diagram depicting an exemplary hybrid auction. The processstarts with step 102, in which memory locations of a computer areinitialized. In one preferred embodiment, the appropriate memorylocations of the bidding information processor (auction computer) areinitialized with information such as the types of items in the auction,the available quantity of each type of item in the auction, an initialprice parameter, an auction timetable, a list of bidder ID's, and a listof passwords. In step 104, a computer implements the earlier phase ofthe auction, which is often a dynamic auction. In one preferredembodiment, the earlier phase of the auction is a clock auction, asshown in greater detail in FIG. 5, below. After the earlier phase of theauction concludes, a computer proceeds to step 106, in which it carriesforward all or part of the bidding history from the earlier phase of theauction to the later phase of the auction. In step 108, a computerimplements the later phase of the auction, which is a package auctiondifferent from the earlier phase of the auction. In one preferredembodiment, the later phase of the auction is a proxy auction, as shownin detail in FIG. 13. In another preferred embodiment, the later phaseof the auction is a sealed bid package auction, as shown in greaterdetail in FIGS. 5, 11 and 16, below. (Observe that an auction processwith mandatory proxy bidding—and in which a bidder is not allowed tomake changes to the flexible bid information that is used by its proxyagent—fits the description of both a proxy auction and of a sealed bidpackage auction.) After the later phase of the auction concludes, acomputer proceeds to step 110, in which a computer outputs a finalmessage that includes the outcome of the later phase of the auction. Inmany preferred embodiments, the outcome of the later phase of theauction also serves as the outcome of the overall hybrid auction. Theprocess then concludes.

FIG. 5 is a flow diagram of a preferred embodiment of the presentinvention, in which there are two phases to the auction: a clock auctionphase; followed by a sealed bid package auction phase. The processstarts with step 112, in which memory locations of a computer areinitialized. In one preferred embodiment, the appropriate memorylocations of the bidding information processor (auction computer) areinitialized with information such as the types of items in the auction,the available quantity of each type of item in the auction, an initialprice parameter, an auction timetable, a list of bidder ID's, and a listof passwords. In step 114, a computer establishes the initial pricevector (P₁, . . . , P_(m)). The process proceeds to step 116, in which acomputer outputs auction information, including the current price vector(P₁, . . . , P_(m)). In one preferred embodiment, the biddinginformation processor outputs the auction information through itsnetwork interface and transmits it via the network. The bidder terminalsthen receive the auction information through their network interfacesand display the information to bidders through their user interfaces. Instep 118, a computer receives quantity vectors (Q₁ ^(i), . . . , Q_(m)^(i)) from bidders. In one preferred embodiment, a bidder inputs hisbids through the user interface of the bidder terminal, which thenoutputs the auction information through its network interface andtransmits it via the network. The bidding information processor thenreceives the bids through its network interface for use in the nextstep. In step 120, a computer applies constraints, if any, to thereceived quantity vectors, and enters as bids only those that satisfysaid constraints. This process is illustrated in greater detail in FIGS.6 a and 6 b. In one preferred embodiment, the constraints are applied atthe bidding information processor, although they may also easily beapplied at the bidder terminals. In step 122, a computer processes theentered bids and determines whether the clock phase of the auctionshould continue. Exemplary processes of step 122 are illustrated ingreater detail in FIG. 10. In some preferred embodiments, thisdetermination occurs at the bidding information processor.

If the clock phase of the auction should continue, the process goes tostep 124, in which a computer establishes an updated price vector (P₁, .. . , P_(m)). Then, at step 126, a computer updates other auctioninformation, if any. In one preferred embodiment, the biddinginformation processor automatically generates a suggested revised pricevector, outputs the suggested revised price vector through its networkinterface, and transmits it via the network. The auctioneer terminalthen receives the suggested revised price vector through its networkinterface and displays it to the auctioneer through its user interface.The auctioneer either approves or modifies the revised price vectorthrough the user interface of the auctioneer terminal, which thenoutputs the revised price vector through its network interface andtransmits it via the network. The bidding information processor thenreceives the revised price vector through its network interface for usein subsequent steps. The process then loops to step 116.

If the clock phase of the auction should not continue, the process goesto step 128, in which a computer initiates a sealed bid phase of theauction. Then, at step 130, a computer receives sealed bids (S^(i),P^(i)) from bidders. The pair (S^(i), P^(i)) is a package bid, whereS^(i)⊂Ω is a package, i.e., a subset of the set of all items beingauctioned, and P^(i) is a price at which bidder i is offering totransact for the entire package S^(i). In step 132, a computer appliesconstraints, if any, to the received sealed bids, and enters only thosesealed bids that satisfy said constraints. This process is illustratedin greater detail in FIG. 10. In one preferred embodiment, theconstraints are applied at the bidding information processor, althoughthey may also easily be applied at the bidder terminals. Following theentering of the sealed bids, the process goes to step 134, in which acomputer determines an allocation of items and payments of bidders,based on bids received in the clock phase of the auction and on thereceived sealed bids. Exemplary processes of step 134 are illustrated ingreater detail in FIGS. 11 and 16. In some preferred embodiments, thisdetermination occurs at the bidding information processor. In someembodiments, the determination may be based only on the received sealedbids (although the bids received in the clock phase could still have theeffect of constraining the sealed bids in step 132). In step 136, acomputer outputs a final message, which includes the allocation andpayment outcome, through its network interface and transmits it via thenetwork. The bidder terminals and auctioneer terminal then receive theallocation and payment outcome through their network interfaces anddisplay the information to bidders and the auctioneer through their userinterfaces. The process then ends.

Elements of the Invention Concerned with Applying Constraints to Bids

In some preferred embodiments of the invention, a revealed-preferenceactivity rule (also known as a “revealed-preference-based constraint”)is imposed on bidders in the earlier phase (e.g. dynamic auction) andthe later phase (e.g. package auction) of the auction. For a longerdiscussion of revealed preference, see the nearly last sections of“System and Method for a Dynamic Auction with Package Bidding,”International Patent Application No. US01/43838.

Here is how they are imposed, in one preferred embodiment. Suppose thatthere are m types of items (1, . . . , m) being auctioned, and let s andt be two times in the auction (s<t) at which bids may be received frombidders. Let price vector P^(s)≡(P₁ ^(s), . . . , P_(m) ^(s)) denote theprices (per unit) for the respective types of items at time s, and letP^(t)≡(P₁ ^(t), . . . , P_(m) ^(t)) denote the prices (per unit) for therespective types of items at time t. Further, for a given bidder i, letquantity vector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) denote thequantities of the respective types of items demanded by bidder i at times, and let Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) denote thequantities of the respective types of items demanded by bidder i at timet.

The revealed-preference activity rule accepts the quantity vectorQ^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) as a bid (together with theimplied price) only if the following inequality holds:(P ^(t) −P ^(s))·(Q ^(i,t) −Q ^(i,s))≦0, for all s<t.  (RP)The relaxed revealed-preference activity rule accepts the quantityvector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) as a bid (together withthe implied price) only if the following inequality holds:(P ^(t) −P ^(s))·(Q ^(i,t) −αQ ^(i,s))≦0, for all s<t.  (RRP)The value α satisfies α≧1. If α>1, this means that the RRP activity ruleis a strictly looser constraint than the RP activity rule. Relaxing theactivity rule is particularly useful in the later phase (e.g. packageauction), for the purpose of reducing collusion in the hybrid auction.

These inequalities may also be expressed in terms of the price for theentire package. Consider two times s and t (s<t). Suppose the bidderbids for the package S at time s and T at time t. Let P^(s)(S) andP^(s)(T) be the package price of S and T at time s; let P^(t)(S) andP^(t)(T) be the package price of S and T at time t; and let ν(S) andν(T) be the value of package S and T. Revealed preference says that thebidder prefers S to T at time s:ν(S)−P ^(s)(S)≧ν(T)−P ^(s)(T),and prefers T to S at time t:ν(T)−P ^(t)(T)≧ν(S)−P ^(t)(S).Adding these two inequalities yields the revealed preference activityrule for packages:P ^(t)(S)−P ^(s)(S)≧P ^(t)(T)−P ^(s)(T).  (RP′)Intuitively, the package price of S must have increased more than thepackage price of T from time s to time t, for otherwise, at time t, Swould be more profitable than T.

Similarly, a relaxed revealed-preference activity rule can be expressedin terms of the price for the entire package:α[P ^(t)(S)−P ^(s)(S)]≧P ^(t)(T)−P ^(s)(T).  (RRP′)As above, the value α satisfies α≧1. Relaxing the activity rule (α>1) isparticularly useful in the later phase of the auction. Either of twoviews may be taken. First, constraint (RRP′) is imposed on thecollection of package bids (T, P^(t)(T)) that a given bidder submits inthe later phase of the auction, that is, the bidder's submission ofpackage bids is entered only if it satisfies constraint (RRP′) relativeto all bids entered for the bidder earlier in the auction.Alternatively, the later phase of the auction is a proxy auction, andthe bidder may submit any valuations into the proxy agent. However, theproxy agent is bound by constraint (RRP′), and it is permitted to bid onthe package T only if (RRP′) is satisfied relative to all bids enteredfor the bidder earlier in the auction.

FIGS. 6 a and 6 b are flow diagrams of two exemplary subprocesses ofstep 120 of FIG. 5. The process of FIG. 6 a begins with step 120 a-1, inwhich a bidder i who has not yet been considered is selected. In step120 a-2, a quantity vector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) forbidder i which has not yet been considered is selected. The quantityvector Q^(i,t)≡(Q^(i,t), . . . , Q_(m) ^(i,t)) was submitted at time tin the auction. In step 120 a-3, it is checked whether each quantityQ_(k) ^(i,t) in the selected vector Q^(i,t) is a nonnegative integer. Ifeach component of the vector is a nonnegative integer, the process goesto step 120 a-4. In step 120 a-4, it is checked whether the selectedquantity vector Q^(i,t) is consistent with bidder i's initialeligibility, that is, whether:

${{\sum\limits_{k = 1}^{m}\;{C_{k}^{i,0}Q_{k}^{i,t}}} \leq {\overset{\_}{E}}^{i,0}},$where bidder i's initial eligibility, Ē^(i,0), may, for example, bedetermined by the level of financial guarantee posted by bidder i. Ifthe selected quantity vector Q^(i,t) is consistent with bidder i'sinitial eligibility, the process goes to step 120 a-5, where bidder i'sprior quantity vectors and associated price vectors, {(Q^(i,s),P^(s))}_(s<t), are recalled. The time s is any time prior to time t(that is, s<t) in the auction. The price vector P^(s)≡(P₁ ^(s), . . . ,P_(m) ^(s)) denotes the current price vector that was outputted in step116 at time s. The quantity vector Q^(i,s)≡(Q₁ ^(i,s), . . . , Q_(m)^(i,s)) identifies a package of items that bidder i was offering totransact at price vector P^(s), and which was entered as a validquantity vector at time s. The process then goes to step 120 a-6, whereit is checked whether the selected quantity vector Q^(i,t) is consistentwith the revealed-preference activity rule, that is, whether theinequality:

${{\sum\limits_{k = 1}^{m}\;{\left( {P_{k}^{t} - P_{k}^{s}} \right)\left( {Q_{k}^{i,t} - Q_{k}^{t,s}} \right)}} \leq 0},$is satisfied for all s<t. [Note that the above inequality is equivalentto inequality (RP), (P^(t)−P^(s))·(Q^(i,t)−Q^(i,s))≦0, for all s<t, butinequality (RP) is written in vector notation.] If it is, the processcontinues to step 120 a-7, where the selected quantity vector Q^(i,t) isentered as a valid quantity vector for bidder i at time t. Optionally,bidder i is sent a message confirming to him that the selected quantityvector Q^(i,t) is valid. The process then goes to step 120 a-8, where itis determined whether all quantity vectors for bidder i at time t havebeen considered. If not, the process loops back to step 120 a-2. If allquantity vectors for bidder i at time t have been considered, theprocess continues to step 120 a-9, where it is determined whether allbidders have been considered. If not, the process loops back to step 120a-1. If all bidders have been considered, the process goes to step 122of FIG. 5.

If the selected quantity vector Q^(i,t) fails any of the checks at steps120 a-3, 120 a-4 or 120 a-6, the process instead goes to step 120 a-10,where a message is outputted that the selected quantity vector Q^(i,t)is invalid. The selected quantity vector then is not entered as part ofa valid bid. The process then goes to step 120 a-8, where it isdetermined whether all quantity vectors for bidder i have beenconsidered. If not, the process loops back to step 120 a-2. If allquantity vectors for bidder i have been considered, the processcontinues to step 120 a-9, where it is determined whether all biddershave been considered. If not, the process loops back to step 120 a-1. Ifall bidders have been considered, the process goes to step 122 of FIG.5.

FIG. 6 b depicts a similar process as FIG. 6 a. However, in the processof FIG. 6 b, quantity vectors are entered for a group G of types ofitems (that is, G⊂{1, . . . , m}), rather than for all types of items(that is, {1, . . . , m}). The process of FIG. 6 b begins with step 120b-1, in which a bidder i who has not yet been considered is selected. Instep 120 b-2, a quantity vector (Q_(k) ^(i,t))_(kεG) for bidder i whichhas not yet been considered is selected. The quantity vector (Q_(k)^(i,t))_(kεG) is a bid for the types of items contained in group G, thatwas submitted at time t in the auction. In step 120 b-3, it is checkedwhether the selected quantity vector (Q_(k) ^(i,t))_(kεG) satisfies theconstraint:

${{\sum\limits_{k \in G}{C_{k}^{i,t}Q_{k}^{i,t}}} \leq {\overset{\_}{C}}_{G}^{i,t}},$where C_(k) ^(i,t) and C _(G) ^(i,t) are arbitrary constants. If theconstraint of step 120 b-3 is satisfied, the process goes to step 120b-4. In step 120 b-4, it is checked whether the selected quantity vector(Q_(k) ^(i,t))_(kεG) satisfies the constraint:

${{\sum\limits_{k \in G}{C_{k}^{{\prime\; i},t}Q_{k}^{i,t}}} \geq {\hat{C}}_{G}^{i,t}},$where C′_(k) ^(i,t) and Ĉ_(G) ^(i,t) are arbitrary constants. If theconstraint of step 120 b-4 is satisfied, the process goes to step 120b-5, where it is checked whether the selected quantity vector (Q_(k)^(i,t))_(kεG) was submitted at a time no earlier than the starting timeof the current round. If it was, the process goes to step 120 b-6, whereit is checked whether the selected bid was submitted at a time no laterthan the ending time of the current round. If it was, the processcontinues to step 120 b-7, where the selected quantity vector (Q_(k)^(i,t))_(kεG) is entered as a valid quantity vector on group G forbidder i at time t. Optionally, bidder i is sent a message confirming tohim that the selected quantity vector is valid. The process then goes tostep 120 b-8, where it is determined whether all quantity vectors forbidder i have been considered. If not, the process loops back to step120 b-2. If all quantity vectors for bidder i have been considered, theprocess continues to step 120 b-9, where it is determined whether allbidders have been considered. If not, the process loops back to step 120b-1. If all bidders have been considered, the process goes to step 122of FIG. 5.

If the selected quantity vector (Q_(k) ^(i,t))_(kεG) fails any of thechecks at steps 120 b-3, 120 b-4, 120 b-5 or 120 b-6, the processinstead goes to step 120 b-l0, where a message is outputted that theselected bid is invalid. The selected quantity vector then is notentered as part of a valid bid. The process then goes to step 120 b-8,where it is determined whether all quantity vectors by bidder i havebeen considered. If not, the process loops back to step 120 b-2. If allquantity vectors by bidder i have been considered, the process continuesto step 120 b-9, where it is determined whether all bidders have beenconsidered. If not, the process loops back to step 120 b-1. If allbidders have been considered, the process goes to step 122 of FIG. 5.

It is important to note that, in many preferred embodiments of the clockauction phase, bidders are allowed full flexibility in making bidswhich, if accepted, would cause aggregate demand to be less than supply.After each new price vector is announced, bidders can arbitrarily reducetheir previous quantities bid. (However, note that the previous bidswill be carried forward to the proxy auction phase in many preferredembodiments, so the bids retain meaning.) For example, it might be thecase that supply equals demand for a particular item, but a bidder maywish to reduce his demand on that item, as the price of a complementaryitem has increased. Or it might be the case that, when demand wasgreater than supply for a particular item, two bidders simultaneouslyattempted to reduce their demands, sufficiently to now make demand lessthan supply. It is tempting to refuse to allow the reduction in thefirst case, or to ration the bidders in the second case, since otherwisethe clock auction phase may yield a significant underselling of theitems in the auction. However, to refuse the reduction or to ration thebidders may yield an exposure problem for bidders who have complementspreferences. Consequently, in many preferred embodiments of the clockauction phase, the full flexibility to reduce arbitrarily reduce bids isallowed. In any event, observe that the clock auction phase does notconclude the auction, and the underselling can be remedied during thelater phase (e.g. package auction phase, including sealed bid auctionphase or proxy auction phase) of the auction.

Embodiments of the Invention Concerned with Whether the Dynamic AuctionPhase Should Continue

FIG. 7 a is a flow diagram of a subprocess of step 122 of FIG. 5. Itillustrates a first exemplary process by which a computer may determinewhether the dynamic auction phase of a hybrid auction should continue.(In particular, FIG. 7 a will illustrate an exemplary process by which acomputer may determine whether the clock auction phase of a two-phaseauction comprising a clock auction phase and a sealed bid phase shouldcontinue. Related to this will also be FIG. 9 b, below, whichillustrates an exemplary process by which a computer determines whetherthe clock auction phase should continue, in a system where bidders arepermitted to submit Intra-Round Bids.)

The process of FIG. 7 a treats a clock auction phase in which eachbidder i is permitted to submit only a single quantity vector associatedwith a current price vector P^(t)≡(P₁ ^(t), . . . , P_(m) ^(t)). Thequantity vector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) identifies apackage of items that bidder i is offering to transact at price vectorP^(t). FIG. 7 a begins with step 122 a-1, in which a type k of items notyet considered is selected. In step 122 a-2, a computer determineswhether the sum of the quantities bid for items of type k (summed overall bidders i=1, . . . , n) is less than or equal to the availablequantity of items of type k, that is, whether:

${\sum\limits_{i = 1}^{n}\; Q_{k}^{i,t}} \leq {{\overset{\_}{Q}}_{k}.}$If this inequality is not satisfied, then type k of items has not yetcleared, and so the dynamic auction phase should continue. The processthus jumps immediately to step 124 of FIG. 5.

If the inequality of step 122 a-2 is satisfied, the process then goes tostep 122 a-3, where it is determined whether all types k of items (k=1,. . . , m) have been considered. If not, the process loops back to step122 a-1. However, if all types k of items have already been considered,then it has been found that all types k of items have cleared, and sothe dynamic auction phase should not continue. The process proceeds tostep 128 of FIG. 5, where a computer initiates the sealed bid phase ofthe hybrid auction.

FIG. 7 b is a flow diagram of a subprocess of step 122 of FIG. 5. Itillustrates a second exemplary process by which a computer may determinewhether the dynamic auction phase of a hybrid auction should continue.(In particular, FIG. 7 b will illustrate an exemplary process by which acomputer may determine whether the clock auction phase of a two-phaseauction comprising a clock auction phase and a sealed bid phase shouldcontinue.)

In the process of FIG. 7 b, market clearing is defined for a group G oftypes of items (that is, G⊂{1, . . . , m}), rather than for everyindividual type of item. FIG. 7 b begins with step 122 b-1, in which agroup G of types of items not yet considered is selected. In step 122b-2, a computer determines whether the excess demand for group G oftypes of items is within C ^(G) of the available quantity, that is,whether:

${{{\overset{\_}{Q}}^{G} - {\sum\limits_{i = 1}^{n}{\sum\limits_{k \in G}Q_{k}^{i,t}}}}} \leq {{\overset{\_}{C}}^{G}.}$The nonnegative constant, C ^(G), has the interpretation that this isthe tolerance to which the auctioneer is allowing oversell or undersellto occur. If the auctioneer needs to sell exactly the available quantityof the group G of item types, then C ^(G)=0. If this inequality is notsatisfied, then group G of item types has not yet cleared, and so thedynamic auction phase should continue. The process thus jumpsimmediately to step 124 of FIG. 5.

If the inequality of step 122 b-2 is satisfied, the process then goes tostep 122 b-3, where it is determined whether all groups G of types ofitems have been considered. If not, the process loops back to step 122b-1. However, if all groups G of types of items have already beenconsidered, then it has been found that all groups G of types of itemshave cleared within a tolerance of C ^(G), and so the dynamic auctionphase should not continue. The process proceeds to step 128 of FIG. 5,where a computer initiates the sealed bid phase of the hybrid auction.

FIG. 7 c is a flow diagram of a subprocess of step 122 of FIG. 5. Itillustrates a third exemplary process by which a computer may determinewhether the dynamic auction phase of a hybrid auction should continue.(In particular, FIG. 7 c will illustrate an exemplary process by which acomputer may determine whether the clock auction phase of a two-phaseauction comprising a clock auction phase and a sealed bid phase shouldcontinue.)

The process of FIG. 7 c treats a clock auction phase in which eachbidder i is permitted to submit multiple quantity vectors associatedwith a current price vector P^(t)≡(P₁ ^(t), . . . , P_(m) ^(t)). Thequantity vector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) identifies apackage of items that bidder i is offering to transact at price vectorP^(t). If, in addition, bidder i submitted a second quantity vectorR^(i,t)≡(R₁ ^(i,t), . . . , R_(m) ^(i,t)) at time t, then R^(i,t)identifies a second package of items that bidder i is offering totransact at price vector P^(t). In many preferred embodiments of theauction process, bidder i may win package Q^(i,t) or package R^(i,t)—butnot both packages. Described differently, the bids for packages Q^(i,t)and R^(i,t) are treated as mutually exclusive. The “mutually exclusive”interpretation of bids does not prevent bidder i from indicating hiswillingness to transact both quantity vectors; all that bidder i wouldneed to do is also submit a quantity vector of Q^(i,t)+R^(i,t).

FIG. 7 c begins with step 122 c-1, in which, for each bidder i (i=1, . .. , n), a computer recalls the entire set S^(i,t) of quantity vectorsreceived from bidder i at the current price vector P^(t). That is,S^(i,t)≡{Q^(i,t): quantity vector Q^(i,t) was received from bidder i attime t}. In step 122 c-2, a computer determines whether there exists aselection of quantity vectors from S^(i,t), one for each bidder, whichcan be satisfied with the available quantity. More precisely:

-   -   Does there exist {Q^(i,t)}_(i=1, . . . , n) such that

${{\sum\limits_{i = 1}^{n}\; Q_{k}^{i}} \leq {\overset{\_}{Q}}_{k}},$for all k=1, . . . , m, and Q^(i,t)εS^(i,t), for all i =1, . . . , n?If there does not exist any selection of quantity vectors from S^(i,t),one for each bidder, which can be satisfied with the available quantity,then the market cannot be cleared at the current price vector P^(t)≡(P₁^(t), . . . , P_(m) ^(t)), and so the dynamic auction phase shouldcontinue. The process thus proceeds to step 124 of FIG. 5.

However, if there does exist a selection of quantity vectors fromS^(i,t), one for each bidder, which can be satisfied with the availablequantity, then it has been found that the market can be cleared at thecurrent price vector P^(i)≡(P₁ ^(i), . . . , P_(m) ^(i)), and so thedynamic auction phase should not continue. The process thus proceeds tostep 128 of FIG. 5, where a computer initiates the sealed bid phase ofthe hybrid auction.

Embodiments of the Invention Concerned with Updating Prices in theDynamic Auction Phase

FIG. 8 a is a flow diagram of a subprocess of step 124 of FIG. 5. Itillustrates a first exemplary process by which a computer may updateprices in the dynamic auction phase. (In particular, FIG. 8 a willillustrate an exemplary process by which a computer may establish anupdated price vector in the clock auction phase of a two-phase auctioncomprising a clock auction phase and a sealed bid phase.)

The process of FIG. 8 a treats a clock auction phase in which eachbidder i is permitted to submit only a single quantity vector associatedwith a current price vector P^(t)≡(P₁ ^(t), . . . , P_(m) ^(t)). Thequantity vector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) identifies apackage of items that bidder i is offering to transact at price vectorP^(t). FIG. 8 a begins with step 124 a-1, in which a computer calculatesthe excess demand, Z_(k) ^(t), for all types k of items (k=1, . . . , m)at time t in the clock auction phase, and recalls prior excess demands,Z_(k) ^(t−1), Z_(k) ^(t−2), Z_(k) ^(t−3), . . . as needed. The excessdemand, Z_(k) ^(t), for the type k of item at time t is defined by:

$Z_{k}^{t} = {{- {\overset{\_}{Q}}_{k}} + {\sum\limits_{i = 1}^{n}\;{Q_{k}^{i,t}.}}}$That is, the vector of excess demands is the amount by which the sum ofthe quantity vectors (summed over all bidders i=1, . . . , n) exceedsthe vector Q of available quantities. The process then goes to step 124a-2, in which a computer calculates price increments, Δ_(k) ^(t), forall types k of items (k=1, . . . , m) at time t in the clock auctionphase. In general, the price increments may be any arbitrary function ofthe state of the auction information. However, in many embodiments ofthe inventive system, the price increments are calculated from theexcess demands. In one preferred embodiment, the price increment foreach type of item is calculated by taking a weighted sum of current andpast excess demands for this item type and other nearby item types. Forexample:Δ_(k) ^(t) =C ₀₀ Z _(k) ^(t) +C ₁₀ (Z _(k+1) ^(t) +Z _(k−1) ^(t))+C ₂₀(Z _(k+2) ^(t) +Z _(k−2) ^(t))+C ₀₁ Z _(k) ^(t−1) +C ₁₁(Z _(k+1) ^(t−1)+Z _(k−1) ^(t−1))+C ₂₁(Z _(k+2) ^(t−1) +Z _(k−2) ^(t−1)where C₀₀, C₁₀, C₂₀, C₀₁, C₁₁ and C₂₁ are positive constants. Oneexemplary application of this price increment formula is for airportslots. The types k of items may refer to landing slots in consecutive15-minute intervals. Thus, landing slots of types k−1 and k+1 are likelyto be reasonable substitutes for landing slots of type k, andconsequently the price increment for landing slots of type k mightdepend, in part, on the excess demand for landing slots of types k−1 andk+1. At the same time, the excess demands at preceding times in theclock auction phase are also relevant information for establishingupdated prices, so the price increment for landing slots at time t mightdepend, in part, on the excess demand for landing slots at time t−1.

The process then goes to step 124 a-3, where the updated price vector isestablished by setting P_(k) ^(t+1)=P_(k) ^(t)+Δ_(k) ^(t), for all typesk of items (k=1, . . . , m). The process then proceeds to step 126 ofFIG. 5, where a computer updates other auction information, if any.

FIG. 8 b is a flow diagram of a subprocess of step 124 of FIG. 5. Itillustrates a second exemplary process by which a computer may updateprices in the dynamic auction phase. (In particular, FIG. 8 b willillustrate an exemplary process by which a computer may establish anupdated price vector in the clock auction phase of a two-phase auctioncomprising a clock auction phase and a sealed bid phase.)

The process of FIG. 8 b treats a clock auction phase in which eachbidder i is permitted to submit multiple quantity vectors associatedwith a current price vector P^(t)≡(P₁ ^(t), . . . , P_(m) ^(t)). Thequantity vector Q^(i,t)≡(Q₁ ^(i,t), . . . , Q_(m) ^(i,t)) identifies apackage of items that bidder i is offering to transact at price vectorP^(t). If, in addition, bidder i submitted a second quantity vectorR^(i,t)≡(R₁ ^(i,t), . . . , R_(m) ^(i,t)) at time t, then R^(i,t)identifies a second package of items that bidder i is offering totransact at price vector P^(t). In many preferred embodiments of theauction process, bidder i may win package Q^(i,t) or package R^(i,t)—butnot both packages. Described differently, the bids for packages Q^(i,t)and R^(i,t) are treated as mutually exclusive. The “mutually exclusive”interpretation of bids does not prevent bidder i from indicating hiswillingness to transact both quantity vectors; all that bidder i wouldneed to do is also submit a quantity vector of Q^(i, t)+R^(i,t).

FIG. 8 b begins with step 124 b-1, in which, for each bidder i (i=1, . .. , n), a computer recalls the entire set S^(i,t) of quantity vectorsreceived from bidder i at the current price vector P^(t). That is,S^(i,t)≡{Q^(i,t): quantity vector Q^(i,t) was received from bidder i attime t}. In step 124 b-2, a computer determines a selection of quantityvectors for each bidder. In one preferred embodiment, a computerdetermines a selection of quantity vectors {tilde over (Q)}^(i,t) fromS^(i,t), one for each bidder, which minimizes the extent to which thesum of the quantity vectors (summed over all bidders i =1, . . . , n)exceeds the vector of available quantities. More precisely, a computerdetermines a solution to the following optimization problem:

-   -   Determine {{tilde over (Q)}^(i,t)}_(i=1, . . . , n) that        minimizes

${\sum\limits_{k = 1}^{m}\;{\max\left\{ {0,{{- {\overset{\_}{Q}}_{k}} + {\sum\limits_{i = 1}^{n}\; Q_{k}^{i,t}}}} \right\}}},$subject to Q^(i,t)εS^(i,t), for all i=1, . . . , n.The process then goes to step 124 b-3, in which a computer calculatesthe excess demands {tilde over (Z)}_(k) ^(t) for all types k of items(k=1, . . . , m) based on the selection {tilde over (Q)}^(i,t) fromS^(i,t) determined in step 124 b-2. The excess demand, {tilde over(Z)}_(k) ^(t), for the type k of item at time t is defined by:

${\overset{\sim}{Z}}_{k}^{t} = {{- {\overset{\_}{Q}}_{k}} + {\sum\limits_{i = 1}^{n}\;{{\overset{\sim}{Q}}_{k}^{i,t}.}}}$That is, the vector of excess demands is the amount by which the sum ofthe quantity vectors {tilde over (Q)}^(i,t) (summed over all biddersi=1, . . . , n) exceeds the vector Q of available quantities. Theprocess then continues to step 124 b-4, in which a computer calculatesprice increments, Δ_(k) ^(t), for all types k of items (k=1, . . . , m)at time t in the clock auction phase. In general, the price incrementsmay be any arbitrary function of the state of the auction information.However, in many embodiments of the inventive system, the priceincrements are calculated from the excess demands. In one preferredembodiment, the price increment for each type of item is calculated bytaking a weighted sum of current excess demands for this item type andother nearby item types. For example:Δ_(k) ^(t) =C ₀₀ {tilde over (Z)} _(k) ^(t) +C ₁₀({tilde over (Z)}_(k+1) ^(t) +{tilde over (Z)} _(k−1) ^(t))+C ₂₀({tilde over (Z)} _(k+2)^(t) +{tilde over (Z)} _(k−2) ^(t))+C ₃₀({tilde over (Z)} _(k+3) ^(t)+{tilde over (Z)} _(k−3) ^(t)),where C₀₀, C₁₀, C₂₀ and C₃₀ are positive constants. As before, oneexemplary application of this price increment formula is for airportslots.

The process then goes to step 124 b-5, where the updated price vector isestablished by setting P_(k) ^(t+1)=P_(k) ^(t)+Δ_(k) ^(t), for all typesk of items (k=1, . . . , m). The process then proceeds to step 126 ofFIG. 5, where a computer updates other auction information, if any.

Embodiments of the Invention Concerned with Intra-Round Bids

In many of the leading dynamic electronic auctions in the prior art,bidders submit bids in a sequence of discrete rounds. For example, inthe Federal Communications Commission auctions for radio communicationsspectrum or in the UMTS (3G) spectrum auctions held by European nations,the following would be a typical bidding schedule for an auction:

-   -   Round 1: 9:00-9:45    -   Round 2: 10:00-10:45    -   Round 3: 11:00-11:45    -   Round 4: 12:00-12:45    -   Round 5: 13:00-13:45    -   Round 6: 14:00-14:45    -   Round 7: 15:00-15:45    -   Round 8: 16:00-16:45        This bidding schedule would have the following interpretation.        During the specified time period of each round, a bidder would        be required to submit a new bid or new collection of bids        (unless this bidder was already the standing high bidder on an        item after the bidding of the previous round). If a bidder who        was required to submit a new bid failed to submit a new bid,        then (except for provisions in the rules concerning automatic        waivers) the bidder would be eliminated from the auction.

By contrast, some other electronic auctions in the prior art—forexample, online auctions at eBay—allow bidding to occur continuously.Rather than adhering to any rigid round schedule, bidders may submitbids at any times that they like up to a specified closing time. Relatedto this, there is no sense that a bidder is required to bid a certainamount by any particular time in order to retain eligibility to bid at alater time in the auction.

Many or most electronic auctions for high-valued items utilize adiscrete round structure, rather than allowing bidding to occurcontinuously. There appear to be several reasons for this. First, adiscrete round structure has desirable information properties. Theauction can be easily structured so that the results of Round t aredisseminated to bidders before the bids of Round t+1 need to besubmitted. Second, a discrete round structure is especially conducive toenforcing “activity rules,” in which a bidder is required to be active(i.e., either be the standing high bidder or place a new high bid) on agiven number of items in an earlier round of the auction in order tocontinue to bid on a given number of items in a later round of theauction. This forces bidders to effectively disclose to their opponents(through their bidding) the values that they attach to the items,helping to mitigate the well-known “Winner's Curse” present in auctions.Third, a discrete round structure requires a bidder to repeatedlyaffirm, in successive rounds, his willingness to pay a given price foran item in the auction—which may be especially desirable when items suchas communications licenses may sell for millions or billions of dollarsor euros.

At the same time, the desirable properties of a discrete round structuremay come at some considerable cost. It will typically be reasonable tohold only something like 8 to 12 rounds of bidding in a given day. As aresult, the auctioneer must accept at least one of several problems:

-   -   (1) The auction may be required to last a very long time: in        some North American and European spectrum auctions, the bidding        extended more than 20 business days. Such a lengthy auction may        be rather onerous for bidders and for the seller. In particular,        it may discourage bidder participation, causing the seller to        forgo substantial revenues.    -   (2) The bid increment between successive rounds may be required        to be rather substantial: in some North American and European        spectrum auctions, the bid increment between successive rounds        never was allowed to drop below five percent of the previous        bid. It can be argued that a seller suffers an expected revenue        loss that is directly proportional to the minimum bid increment,        so this may cost a seller millions of dollars or euros.    -   (3) The starting price may be required to be very near to the        expected closing price. This may discourage bidder        participation, as well as potentially eliminating the        possibility of bidders getting caught up in the excitement of        the auction and bidding very high prices (which is one of the        advantages of conducting a dynamic auction). This also runs the        risk that the auction will fail: that is, quantities bid at the        starting price being less than the available quantity at the        auction.        Moreover, in a clock auction, problem (2) above, a large bid        increment, may lead to a heightened risk of “undersell”.        Consider an auction with an available quantity of 100 units of        an item, and suppose a bid increment of five percent. It is        quite plausible that, at a price of $1,000,000 per unit, the        aggregate quantity bid by all bidders would equal 110 units, but        at the next price of $1,050,000 per unit, the aggregate quantity        bid by all bidders would decline to only 60 units. The        auctioneer then faces the unattractive alternatives of: selling        only 60 units out of the available quantity of 100 units at a        price of $1,050,000 each; rationing bidders so that only 100        units, out of the 110 demanded, are sold at $1,000,000; or        restarting the auction at $1,000,000. Observe however that the        “undersell” problem would in all likelihood have been        substantially avoided, had a much smaller bid increment been        possible.

One embodiment of the present invention is a system and method for“Intra-Round Bids.” A discrete round structure—with all of its manyadvantages—is preserved for a clock auction. However, in each round ofthe clock auction, a “starting price” and “ending price” is establishedfor each type of item. Bidders are permitted to submit bids at pricesbetween the starting price and the ending price. In a preferredembodiment, a bidder submits a price parameter for group G representinga percentage of the distance from the starting price vector for group Gand the ending price vector for group G.

Bidders have every incentive to utilize Intra-Round Bids, and to theextent that bidders utilize them, the seller should be expected toattain higher auction revenues and to reduce the probability ofundersell. Thus, a system and method for Intra-Round Bids improves uponthe prior art for auction systems and methods, and has immediatepractical application for dynamic auctions of radio communicationsspectrum, securities and other financial products, electric power, etc.

While the previous and following description of Intra-Round Bids isframed largely in terms of regular auctions to sell (where bidders arebuyers), the invention is equally applicable for reverse or procurementauctions to buy (where bidders are sellers). For the sake of brevity,this specification will not run through the process a second time withthe roles of selling and buying reversed, but it should be clear toanybody skilled in the art that the technology can be equally used inboth situations.

Here is an example illustrating the usefulness and exact meaning ofIntra-Round Bids. Suppose that, in a clock auction with an availablequantity of 100 units, the ending price per unit associated with Round 4is $1,000,000, and the ending price per unit associated with Round 5 is$1,050,000. In an auction with discrete bidding rounds, Bidder 1 mightsubmit a bid quantity of 55 units for Round 4 and a bid quantity of 30units for Round 5. If there also exists a Bidder 2 who submits the samebid quantities, then we would have exactly the “undersell” problemdescribed above: an aggregate quantity bid by all bidders of 110 unitsin Round 4but only 60 units in Round 5 (with available quantity of 100units).

With an auction system and method with Intra-Round Bidding, the endingprice for Round 4 may be taken to be the starting price for Round 5,i.e., the starting price for Round 5 is $1,000,000. Here is an exampleof the bids that Bidder 1 might submit for Auction Round 5:

-   -   53 units at $1,010,000 per unit;    -   51 units at $1,020,000 per unit;    -   49 units at $1,030,000 per unit;    -   45 units at $1,035,000 per unit;    -   40 units at $1,040,000 per unit; and    -   30 units at $1,045,000 per unit.

These bids have the following exact meaning: the parameterscorresponding to price indicate the price at which Bidder 1 wishes tochange his quantity demanded as compared to his “previous” (that is,next lower price) bid. Thus, in this example:

-   -   Bidder 1 is willing to purchase 55 units (his previous bid from        Round 4) at prices of $1,000,001-$1,009,999;    -   Bidder 1 is willing to purchase 53 units at prices of        $1,010,000-$1,019,999;    -   Bidder 1 is willing to purchase 51 units at prices of        $1,020,000-$1,029,999;

Bidder 1 is willing to purchase 49 units at prices of$1,030,000-$1,034,999;

Bidder 1 is willing to purchase 45 units at prices of$1,035,000-$1,039,999;

Bidder 1 is willing to purchase 40 units at prices of$1,040,000-$1,044,999; and

Bidder 1 is willing to purchase 30 units at prices of$1,045,000-$1,050,000.

If there also exists a Bidder 2 who submits the same bid quantities,then the auctioneer would be able to declare the auction over at a pricebetween $1,030,000 and $1,034,999, with 98 out of the 100 availableunits sold. The auction revenues are improved, and the undersell problemis greatly reduced.

FIG. 9 a is a flow diagram of a subprocess of step 118 of FIG. 5. Itillustrates an exemplary process by which a particular bidder i maysubmit Intra-Round Bids. FIG. 9 a begins with step 118-1, in whichbidder i selects a group, G, of item types on which he wishes to place abid. In various embodiments of the inventive system, the group, G, maybe the entire set of types k of items (G={1, . . . , m}), or any subsetthereof. In step 118-2, bidder i selects a price parameter for group Grepresenting a percentage of the distance from the starting price vectorfor group G and the ending price vector for group G in the currentround. For example, in a group containing three types of items, if thestarting price vector is (4.00, 4.50, 4.75), if the ending price vectoris (8.00, 8.50, 8.75), and if a bidder enters a price parameter of 25%,this signifies that the bidder is indicating an implied price vector of(5.00, 5.50, 5.75). In step 118-3, bidder i selects quantities of theitem types of group G that he would like to take effect as bids at theprice vector implied by the selected price parameter. In step 118-4,bidder i expresses whether he wishes to enter more bids. If so, theprocess loops back to step 118-1. If not, the process continues to step118-5. In step 118-5, a computer determines whether bidder i hassubmitted at least one bid for each group G of item types. If not, theprocess loops back to step 118-1, and optionally a computer promptsbidder i to submit bids on the groups G of item types on which bidder ihas not submitted at least one valid bid in the current round. If so,the process goes to step 120 of FIG. 5.

FIG. 9 b is a flow diagram of a subprocess of step 122 of FIG. 5. Itillustrates an exemplary process by which a computer determines whetherthe dynamic auction phase should continue, in a system where bidders arepermitted to submit In-tra-Round Bids. FIG. 9 b begins with step 122b-1, in which a group G of item types not yet considered is selected. Instep 122 b-2, a computer sorts all bids entered for group G in thecurrent round. The sorting is done: first, by bidder ID; second, byprice parameter in the entered bid (in descending order); and third, bytime stamp of submission (in descending order). In step 122 b-3, acomputer selects, for each bidder i, the bid, Q^(G,i), for group G withthe highest price parameter (and then the latest time stamp). In step122 b-4, a computer determines whether the aggregate quantity bid forgroup G is no greater than the available quantity, that is, whether:

${\sum\limits_{i = 1}^{n}{\sum\limits_{k \in G}Q_{k}^{G,i}}} \leq {{\overset{\_}{Q}}^{G}.}$If this inequality is not satisfied, then group G of item types has notyet cleared, and so the dynamic auction phase should continue. Theprocess thus jumps immediately to step 124 of FIG. 5.

If the inequality of step 122 b-4 is satisfied, the process then goes tostep 122 b-5, where it is determined whether all groups G of item typeshave been considered. If not, the process loops back to step 122 b-1.However, if all groups G of item types have already been considered,then it has been found that all groups G of item types have cleared, andso the dynamic auction phase should not continue. The process proceedsto step 128 of FIG. 5, where the outcome of the dynamic auction phasemay be generated and a computer initiates the sealed bid phase.

FIG. 9 c is a flow diagram of a subprocess of step 128 of FIG. 5. Itillustrates an exemplary process by which a computer determines theoutcome of the dynamic auction phase, in a system where bidders arepermitted to submit Intra-Round Bids. FIG. 9 c begins with step 128-1,in which for all bids entered in the current (i.e., final) round of thedynamic auction phase, a computer sorts the price parameters fromsmallest to largest, and denotes them π₁<π₂<. . . <π_(R). In step 128-2,a computer initializes the price parameter subscript to r=1, so that inthe first iteration of the remaining steps, the computer considers thesmallest value, π₁. In step 128-3, a group G of item types not yetconsidered is selected. In step 128-4, a computer sorts all bids enteredfor group G in the current round. The sorting is done: first, by bidderID; second, by price parameter in the entered bid (in descending order);and third, by time stamp of submission (in descending order). In step128-5, a computer selects, for each bidder i, the bid, Q^(G,i), forgroup G with the highest price parameter that is less than or equal toπ_(r) (and then with the latest time stamp). In step 128-6, a computerdetermines whether the aggregate quantity bid for group G is no greaterthan the available quantity, that is, whether:

${\sum\limits_{i = 1}^{n}{\sum\limits_{k \in G}Q_{k}^{G,i}}} \leq {{\overset{\_}{Q}}^{G}.}$If this inequality is not satisfied, then group G of item types has notyet cleared at price parameter π_(r), and so r needs to be incremented.The process thus goes to step 128-8, where the price parameter subscriptr is advanced by 1, so that in the next iteration of these steps, thecomputer considers the next price parameter, π_(r+1). The process thenloops back to step 128-3, using the new higher value of π_(r+1) andstarting over for groups G of item types.

If the inequality of step 128-6 is satisfied, the process continues tostep 128-7, where it is determined whether all groups G of item typeshave been considered. If not, the process loops back to step 128-3.However, if all groups G of item types have already been considered,then it has been found that all groups G of item types have cleared atprice parameter π_(r). Thus, the price parameter π_(r) impliesmarket-clearing prices for the dynamic auction phase. The processproceeds to calculate the price vector implied by price parameter π_(r),to note the quantities bid by all bidders at this price vector, and toincorporate these computations into an outcome of the dynamic auctionphase. A computer then initiates the later phase of the auction andproceeds to step 130 of FIG. 5.

Embodiments of the Invention Concerned with the Later Phase of theAuction

Following a determination that the earlier phase of the auction shouldnot continue, the process continues by initiating the later phase of theauction. This is illustrated beginning at step 106 of FIG. 4. Forpreferred embodiments in which the earlier phase is a clock auctionphase and the later phase is a sealed bid auction phase, this isillustrated in greater detail beginning at step 128 of FIG. 5.

FIG. 10 is a flow diagram of an exemplary subprocess of step 132 of FIG.5. The process of FIG. 10 begins with step 132 a-1, in which a bidder iwho has not yet been considered is selected. In step 132 a-2, a computerrecalls the entire set Σ^(i)of bids received from bidder i in the sealedbid phase of the auction. Each bid comprises a pair (S^(i), P^(i)),where S^(i) identifies a set of items and P^(i) identifies an associatedprice. In step 132 a-3, it is checked whether each S^(i)⊂Ω and whethereach P^(i)≧0, that is, whether each S^(i) is a subset of the set of allitems in the auction and whether each P^(i) indicates a nonnegativeassociated price, in other words, whether (S^(i), P^(i)) is a validpackage bid. If each S^(i)⊂Ω and if each P^(i)≧0, the process goes tostep 132 a-4. In step 132 a-4, it is checked whether each Pi isconsistent with bidder i's initial eligibility, that is, whetherP^(i)≦Ē^(i,0), where bidder i's initial eligibility, Ē^(i,0), may, forexample, be determined by the level of financial guarantee posted bybidder i. If each P^(i) is consistent with bidder i's initialeligibility, the process goes to step 132 a-5, where the entirecollection of bidder i's quantity vectors and associated price vectors,{(Q^(i,s), P^(s))}, from the earlier phase of the auction are recalledand converted into package bids. Note that (Q^(i,s), P^(s)) is convertedinto a package bid (Q^(i,s), {circumflex over (P)}^(i,s)) bycalculating:

${\hat{P}}^{i,s} = {\sum\limits_{k = 1}^{m}\;{P_{k}^{s}{Q_{k}^{i,s}.}}}$The difference between P^(s) and {circumflex over (P)}^(i,s) is that{circumflex over (P)}^(i,s) is a scalar-valued price—as is required fora package bid—while P^(s) is an m-dimensional vector.

The process then goes to step 132 a-6, where it is checked whether theset Σ^(i) of bids received from bidder i in the later phase of theauction is consistent with an activity rule applied relative to thecollection of bidder i's quantity vectors and associated price vectors,{(Q^(i,s), P^(s))}, from the earlier phase of the auction. One simpleembodiment of such an activity rule is a monotonicity rule that requireswhenever S^(i) from a bid (S^(i), P^(i)) in the later phase representsthe same set of items as Q^(i,s) from a bid (Q^(i,s), {circumflex over(P)}^(i,s)) in the earlier phase, then P^(i)≧{circumflex over(P)}^(i,s). That is, each bidder i is permitted to only bid a higherprice for each package in the later phase than in the earlier phase. Amore complex embodiment of such an activity rule is the relaxedrevealed-preference activity rule, which checks whether the set Σ^(i) ofbids received from bidder i in the later phase of the auction isconsistent with the constraint (RP′) that was described above. Recallthat revealed preference may be expressed in terms of the price for theentire package. Consider two times s and t (s<t), where s is during theearlier phase and t is during the later phase. Suppose the bidder bidsfor the package S at time s and T at time t. The implied price forpackage T at time s can be calculated by writing package T in thequantity vector notation Q^(i,t), and calculating:

${P^{s}(T)} = {\sum\limits_{k = 1}^{m}\;{P_{k}^{s}{Q_{k}^{i,t}.}}}$The relaxed revealed-preference activity rule may be restated in termsof the price for the entire package:α[P ^(t)(S)−P ^(s)(S)]≧P ^(t)(T)−P ^(s)(T),  (RRP′)where, as above, the value α satisfies α≧1. Relaxing the activity rule(α>1) in the later phase of the auction allows bidders a certain amountof flexibility.

If the set Σ^(i) of bids received from bidder i in the later phase ofthe auction is consistent with an activity rule applied relative to thecollection of bidder i's quantity vectors and associated price vectorsfrom the earlier phase of the auction, the process continues to step 132a-7, where the received set Σ^(i) of bids is entered as a valid set ofpackage bids in the later phase of the auction for bidder i. Optionally,bidder i is sent a message confirming to him that the received set Σ^(i)of bids is valid. The process then continues to step 132 a-8, where itis determined whether all bidders have been considered. If not, theprocess loops back to step 132 a-1 . If all bidders have beenconsidered, the process goes to step 134 of FIG. 5.

If the set Σ^(i) of bids received from bidder i in the later phase ofthe auction fails any of the checks at steps 132 a-3, 132 a-4 or 132a-6, the process instead goes to step 132 a-9, where a message isoutputted that the received set Σ^(i) of bids is invalid. The receivedset Σ^(i) of bids then is not entered as a valid set of package bids inthe later phase of the auction for bidder i. The process then goes tostep 132 a-8, where it is determined whether all bidders have beenconsidered. If not, the process loops back to step 132 a-1. If allbidders have been considered, the process goes to step 134 of FIG. 5.

Winner Determination Problem

After applying constraints to the received sealed bids and entering onlybids that satisfy the constraints, in many preferred embodiments of thepresent invention, a computer determines the allocation of items andpayments of bidders. We define a winner determination problem to be acomputational problem of selecting a combination of winning bids thatoptimizes the revenue, subject to the constraint that the selectedcombination of winning bids is feasible. In some preferred embodiments,a winner determination problem is solved on a computer.

In a standard auction (i.e., an auction to sell), the revenues would bemaximized in a winner determination problem. In a reverse auction (i.e.,a procurement auction), the revenues would be minimized in a winnerdetermination problem. In an auction with unique items, the basicfeasibility constraint in a winner determination problem is theconstraint that each item can be allocated to at most one bidder. In anauction with one or more types of items and an available quantity ofeach type, the basic feasibility constraint in a winner determinationproblem is the constraint that the sum of the quantity vectorsassociated with the winning bids must not exceed the vector of availablequantities. The latter constraint subsumes the former constraint, sincean auction with m unique items can be represented as an auction with mtypes of items, with the available quantity being a vector of 1's.

FIG. 11 is a flow diagram of an exemplary subprocess of step 134 of FIG.5, for an auction with m types of items. The process of FIG. 11 beginswith step 134 a-1, in which a bidder i who has not yet been consideredis selected. In step 134 a-2, a computer recalls the entire set Σ^(i) ofbids that were received from bidder i in the sealed bid phase of theauction and entered in step 132 of FIG. 5. If necessary, each bid(S^(i), P^(i)) in Σ^(i) is converted into the quantity vector notation,(Q^(i), P^(i)), where Q^(i)≡(Q₁ ^(i), . . . , Q_(m) ^(i)) and Q_(k) ^(i)denotes the quantity of items of type k in the set S^(i) of items.Meanwhile, P^(i) identifies a price for the entire package identified byQ^(i). The process continues to step 134 a-3, where the entirecollection of bidder i's quantity vectors and associated price vectors,{(Q^(i,s), P^(s))}, from the earlier phase of the auction are recalledand converted into package bids. Recall that (Q^(i,s), P^(s)) isconverted into a package bid (Q^(i,s), {circumflex over (P)}^(i,s)) bycalculating:

${\hat{P}}^{i,s} = {\sum\limits_{k = 1}^{m}\;{P_{k}^{s}{Q_{k}^{i,s}.}}}$The process then continues to step 134 a-4, where it is determinedwhether all bidders have been considered. If not, the process loops backto step 134 a-1. If all bidders have been considered, the process goesto step 134 a-5, where a computer solves a winner determination problem.

In many preferred embodiments of the present invention, all bids forbidder i in the later phase of the auction and all bids for bidder i inthe earlier phase of the auction are treated as mutually exclusive. Inthat event, and for the case of a standard auction (i.e., an auction tosell), the winner determination problem may be stated:

${Maximize}\mspace{14mu}{\sum\limits_{i = 1}^{n}\;{\overset{\sim}{P}}^{i}}$subject to:

-   -   At most one winning bid ({tilde over (Q)}^(i), {tilde over        (P)}^(i)) is selected for each bidder i =1, . . . , n;    -   If bidder i is a winning bidder, then {tilde over (P)}^(i) is        the price in his winning bid;    -   If bidder i is a losing bidder, then {tilde over (P)}^(i)=0; and

${\sum\limits_{i = 1}^{n}\;{\overset{\sim}{Q}}_{k}^{i}} \leq {{\overset{\_}{Q}}_{k}\mspace{14mu}\text{(the~~feasibility~~constraint).}}$For the case of a reverse auction (i.e., a procurement auction), theword “maximize” in the winner determination problem is replaced by“minimize”.

In some other preferred embodiments of the present invention, bidder i'sbids are not treated as mutually exclusive. In that event, and for thecase of a standard auction (i.e., an auction to sell), the winnerdetermination problem may be stated:

$\text{Maximize~~}\mspace{11mu}{\sum\limits_{i = 1}^{n}\;{\sum\limits_{t = 1}^{T_{i}}\;{\overset{\sim}{P}}^{it}}}$subject to:

-   -   Winning bids ({tilde over (Q)}^(i1), {tilde over (P)}^(i1)), . .        . , ({tilde over (Q)}^(iT) ^(i) , {tilde over (P)}^(iT) ^(i) )        are selected for bidder i;    -   If bidder i is a winning bidder, then {tilde over (P)}^(it) are        the prices in his winning bids;    -   If bidder i is a losing bidder, then {tilde over (P)}^(it) =0;        and

${\sum\limits_{i = 1}^{n}\;{\sum\limits_{t = 1}^{T_{i}}\;{\overset{\sim}{Q}}_{k}^{it}}} \leq {{\overset{\_}{Q}}_{k}\mspace{14mu}\text{(the~~feasibility~~constraint).}}$Again, for the case of a reverse auction (i.e., a procurement auction),the word “maximize” in the winner determination problem is replaced by“minimize”. In many preferred embodiments in which bidder i's bids arenot treated as mutually exclusive, there are nevertheless constraints onwhen multiple bids from bidder i are taken to be winners.

After solving a winner determination problem, the process continues tostep 134 a-6, where the allocation of items and payments of bidders isdetermined. In the above preferred embodiment in which bids were treatedas mutually exclusive, it is determined that the allocation to winningbidder i is the items in the package identified by {tilde over (Q)}^(i)and that the payment of winning bidder i is the price {tilde over(P)}^(i). No items are assigned to a losing bidder and the payment of alosing bidder is zero. In the above preferred embodiment in which bidswere not treated as mutually exclusive, it is determined that theallocation to winning bidder i is the items in the package identified by

$\sum\limits_{t = 1}^{T_{i}}\;{\overset{\sim}{Q}}^{it}$and that the payment of winning bidder i is the price

$\sum\limits_{t = 1}^{T_{i}}\;{{\overset{\sim}{P}}^{it}.}$No items are assigned to a losing bidder and the payment of a losingbidder is zero. After the allocation and payments have been determined,the process goes to step 136 of FIG. 5, where a computer outputs a finalmessage, including the allocation of items and payments of bidders.Proxy Auction as the Later Phase

In many preferred embodiments of the present invention, the later phaseof the hybrid auction is a proxy auction. FIG. 12 is a high-leveldepiction of the architecture of an exemplary auction system in whichbidding is intermediated by proxy agents, and in which changes to theinstructions of proxy agents may be allowed or not allowed, inaccordance with an embodiment of the present invention. In the exemplarygraphical depiction of FIG. 12, the computer system consists of a serverand multiple user computers or terminals. User 30 (the auctioneer)communicates with server 10 (the main auction computer) over a network40. Users 20 a-n (the bidders) also communicate with server 10 over anetwork 40, but all communications from the respective bidders to theauction process are intermediated through the corresponding proxy agents50 a-n. The proxy agents 50 a-n are subsystems of the computer system,and they may physically reside on the bidder computers or terminals 20a-n, the server or auction computer 10, or any other computer.

In FIG. 12, bidders a-n participate in the auction by entering flexiblebid information or making changes in their flexible bid information attheir bidder computers or BT's (20 a-n). The bidders can enter or changetheir flexible bid information at times when the auction system is setto allow changes in the flexible bid information of the respectivebidders. The actual bidding on behalf of the respective bidders isperformed by the proxy agents 50 a-n acting on behalf of the respectivebidders. Based on the respective bidder's flexible bid information, theproxy agent may compute a bid and submit it in the auction process bytransmitting it via a network interface. Meanwhile, the server 10 orauctioneer computer or AT 30 may receive submitted bids, processsubmitted bids, and update the auction state. This is described ingreater detail elsewhere in this application. The server 10 orauctioneer computer or AT 30 may also change the setting of the auctionsystem so as to allow or to not allow bidders to make changes to theirflexible bid information. One exemplary way in which this may be done isthat the server 10 will compute, according to a predetermined rule,whether flexible bid information changes should be allowed and will sendout data to the proxy agents 50 a-n, the bidder computers or BT's 20 a-nand the auctioneer computer or AT 30 indicating whether flexible bidinformation changes are allowed. The proxy agents or bidder computerscarry out the server's instructions on whether flexible bid informationchanges are allowed. Meanwhile, the auctioneer has final authority overwhether flexible bid information changes are allowed, and can overridethe server's determination in this regard, if desired.

The “server” (or auction computer) typically has a central role,especially with regard to communications. In some preferred embodiments,the server also does all of the computations and stores all of the data.In some embodiments the “auctioneer” is a live person who sits down atthe auctioneer terminal, logs in, and makes decisions which affect theconduct of the auction. Decisions that the auctioneer makes includeinitialization decisions necessary to initialize an auction such assetting the size of bid increments that will be used and setting theround schedules. Other decisions include determining the “final call”and calling the end of the auction (both typically based on computationsand a recommendation by the server). Finally the auctioneer can makedecisions in exceptional circumstances such as sending out messages tobidders and placing bids on behalf of bidders whose Internet connectionshave failed. Thus aside from the initialization decisions andexceptional events, the auctioneer's decisions can be no more thanmerely confirming recommendations of other entities. Consequently, inmany embodiments, the auctions may be completely automatic, i.e., withno need for human intervention by an auctioneer.

Flow Diagram of Augmented Dynamic Package-Bidding Auction Process

In dynamic package-bidding auction processes in the prior art, a bidcomprises a package of items and an associated price for the package.That is, bidders merely submit bids comprising pairs, (S, P), where S⊂Ωis a subset of the set of all items being auctioned and P is a price atwhich the bidder is offering to transact for the subset S. There is noscope for bidders to include other information, beyond S and P, in theirbids. Furthermore, the provisional revenues are computed simply byoptimizing an objective function comprising the sum of the prices in theselected bids, subject to a selection constraint that the bids arecompatible (e.g., at most one bid is selected for each item beingauctioned). There is no scope for the auction computer to include, inthe objective function being optimized or in the selection constraintbeing applied, the other information that might be explicitly includedin bids. Nor is there scope for the auction computer to include, in theobjective function being optimized or in the selection constraint beingapplied, bidder-specific attributes that might be implicitly included inbids (via the identity of the qualified bidder submitting a given bid).

The limitations in the prior art, as summarized in the previousparagraph, limit the applicability and usefulness of dynamicpackage-bidding auction processes. Conversely, an “augmented dynamicpackage-bidding auction process,” in which any of the limitationssummarized in the previous paragraph (or combinations thereof) areeliminated, offers a myriad of new and useful applications. An augmenteddynamic package-bidding auction process is thus defined to be anydynamic auction in which package bids are allowed, which includes one ormore of the following features: bidders may include other information,beyond a package of items and an associated price for the package, intheir bids; the auction computer may include, in the objective functionbeing optimized or in the selection constraint being applied, the otherinformation that might be explicitly included in bids; and the auctioncomputer may include, in the objective function being optimized or inthe selection constraint being applied, bidder-specific attributes thatmight be implicitly included in bids (via the identity of the qualifiedbidder submitting a given bid). An augmented dynamic package-biddingauction process may yield efficient outcomes, taking the otherinformation and bidder-specific attributes into account.

The following are some examples of the “other information” that mightexplicitly be included in bids, to useful effect:

-   -   The terms of payment (e.g., cash-on-delivery versus payment in        30 days)    -   The use to which the auctioned items will be put, in a        government auction    -   The quality of the items being provided, in a procurement        auction    -   The delivery times of the items being provided, in a procurement        auction

The following are some examples of the “bidder-specific information”that might implicitly be taken to be included in bids of qualifiedbidders, to useful effect:

-   -   The length of time that the bidder has been in business    -   The credit-rating of the bidder    -   The location of the bidder    -   The status of the bidder as a minority-owned business or a small        business    -   The status of the bidder as a domestic or foreign firm

The following are some examples of how the “other information” or“bidder-specific information” might be included, in the objectivefunction being optimized, to useful effect:

-   -   A higher rating may be assigned to higher-quality items being        provided    -   A higher rating may be assigned to a selection of bids which        includes at least two provisional winners that are        minority-owned businesses or small businesses    -   A higher rating may be assigned to a selection of bids for which        at least 50% of each type of good is available for delivery        within one week

The following are some examples of how the “other information” or“bidder-specific information” might be included, in the selectionconstraint, to useful effect:

-   -   A selection constraint may be applied that at least one-third of        each type of good be provided by an alternate supplier        (second-sourcing)    -   A selection constraint may be applied requiring that at least        two provisional winners be minority-owned businesses or small        businesses    -   A selection constraint may be applied requiring that at least        50% of each type of good be available for delivery within one        week

An augmented dynamic package-bidding auction process may be implementedon a computer in a system with mandatory proxy bidding, according toFIG. 13.

Flow Diagram of Proxy Auction Phase

FIG. 13 is a flow diagram of the later phase of an auction in accordancewith an embodiment of the present invention: a proxy auction, in whichit is mandatory that bidding be intermediated by proxy agents. As such,it provides an illustration of step 108 of FIG. 4. The process startswith step 142, in which memory locations of a computer are initialized.In one preferred embodiment, the appropriate memory locations of theauction server are initialized with information such as the items in theauction, the auction schedule, the minimum opening bids or reserveprices, a list of bidder ID's, a list of passwords, a list ofconstraints on bids, and a list of the bids of each bidder from theclock auction phase. These were carried forward in step 106 of FIG. 4,for use in the proxy auction phase. In step 144, a computer outputs thecurrent auction information (if any) available to bidders, possiblyincluding, for example, the minimum opening bids or current high bids,and whether one or more bidders have been given a “last call” for makingchanges to their flexible bid information. In one preferred embodiment,the auction server outputs the auction information through its networkinterface and transmits it via the network. The user computers orterminals then receive the auction information through their networkinterfaces and display the information to bidders and the auctioneerthrough their user interfaces. In step 146, changes to the flexible bidinformation for given bidders are entered into computer databases ormemory, provided that changes are permitted for the respective bidders(and provided that the bidders wish to make changes to their flexiblebid information). This step is illustrated in greater detail in FIGS. 14a and 14 b. In one preferred embodiment, a bidder inputs his (or her)flexible bid information through the user interface of the biddercomputer or terminal, which then (if necessary) outputs the auctioninformation through its network interface and transmits it via thenetwork. The proxy agent corresponding to that bidder (if located onanother computer) then receives the flexible bid information through itsnetwork interface for use in the next step. In step 148, the proxyagents compute new bids, based on the flexible bid information and thecurrent auction information, to submit on behalf of thier respectivebidders, and the proxy agents submit new bids (if any) in the auctionprocess on behalf of their respective bidders. This step is illustratedin greater detail in FIGS. 15 a and 15 b. In many preferred embodiments,bids comprise pairs (S,P), where S⊂Ω is a subset of the set of all itemsbeing auctioned and P is a price at which the bidder is offering totransact for the subset S. Stated differently, a bid comprises a packageof items and an associated price for the package. As already definedabove, such a bid comprising a pair, (S, P), is defined to be a “packagebid.” In one preferred embodiment, the proxy agents reside on theauction server, so that they can submit new package bids without makinguse of the network. In a second preferred embodiment, the proxy agentsreside on the bidder computers or terminals, in which case the biddercomputers or terminals output the submitted new bids through theirnetwork interfaces and transmit them via the network. The auction serverthen receives the submitted new bids through its network interface foruse in the next step. In step 150, a computer applies constraints, ifany, to the new bids submitted by the proxy agents, and enters onlythose bids that satisfy said constraints. In one preferred embodiment,the constraints are applied at the auction server, although they mayalso easily be applied at the bidder computers or terminals, or at othercomputers.

In step 152, a computer calculates the provisionally-winning bids andprovisional revenues, based on the new bids entered and the previousbids that remain “in effect” (i.e., the previous bids that remainactive, or remain subject to being selected as winning bids). In onepreferred embodiment, the previous bids that remain “in effect” includeall of the bids of each bidder from the clock auction phase. In thispreferred embodiment, all bids take the form of package bids, all bidsthat are entered at any time during the auction remain in effect for theduration of the auction, and all bids that are entered on behalf of agiven bidder are treated as being mutually exclusive. Therefore, in thispreferred embodiment, a computer (which may be the auction server orsome other computer) calculates a solution to the following problem ofoptimizing bid revenues over compatible bids:

-   -   Find an n-type, {(S₁,P₁), . . . , (S_(n),P_(n))}, of bids, one        from each bidder i (i=1, . . . , n), which maximizes the sum        P₁+. . . +P_(n), subject to the constraint that the S_(i) are        disjoint subsets of Ω. Stated differently, for every i (i=1, . .        . , n) and for every j≠i (j=1, . . . , n), it is required that        (S_(i),P_(i)) be a new or previous bid entered by or on behalf        of bidder i, (S_(j),P_(j)) be a new or previous bid entered by        or on behalf of bidder j, and S_(i)∩S_(j)=Ø, i.e. no item of set        S_(i) is a member of the set S_(j)if i≠j.        In performing the above calculation, the computer may take as        implicit the existence of a zero bid, i.e. the pair (└, 0),        associated with each bidder. The calculated n-tuple, {(S₁,P₁), .        . . , (S_(n),P_(n))}, of bids solving the above optimization        problem is defined to be the provisionally-winning bids; and the        calculated sum P₁+. . . +P_(n) is defined to be the provisional        revenues. However, in other preferred embodiments: (a) only some        of the bids that were previously entered into the auction remain        in effect for subsequent calculations of the        provisionally-winning bids; (b) not all bids that are entered on        behalf of a given bidder are treated as being mutually        exclusive, so that the optimization problem may allow two or        more bids by a single bidder to be selected; and (c) the auction        may be an auction to buy, a procurement auction or a reverse        auction (rather than an auction to sell), so that the        optimization problem for calculating provisionally-winning bids        may involve the minimization of payments associated with        selected bids, or some other optimization problem, rather than        the maximization problem stated above. Also, in many preferred        embodiments, a computer stores the calculated        provisionally-winning bids and provisional revenues in memory or        on a data storage device for future use. In step 154, a computer        determines whether the auction should continue. One exemplary        way to perform step 154 is for the auction server to compare the        current provisional revenues with a function of the provisional        revenues obtained in previous iteration(s) of the loop, and to        continue the auction if and only if the current provisional        revenues exceed the function of the provisional revenues        obtained in previous iteration(s). However, this particular        stopping rule is only exemplary, and many other embodiments are        also possible: for example, the rule applied may be different,        it may be performed on a different computer, and the computer        may only produce a recommendation of stopping the auction which        is then transmitted to the auctioneer computer or terminal for        final approval.

If the auction should continue, the process goes to step 156, where itis determined whether one or more bidders should be given a “last call”to change their flexible bid information. The auction server recommendsa decision on whether bidders should be given a “last call” andtransmits this recommendation via the network to the auctioneer computeror terminal. The auctioneer computer or terminal then receives therecommendation through its network interface and displays it to theauctioneer through its user interface. The auctioneer either approves ormodifies the recommendation through the user interface of the auctioneerterminal, which then outputs the final decision through its networkinterface and transmits it via the network. The auction server thenreceives the final decision through its network interface for use insubsequent steps. The process then goes to step 158, in which the stateof the auction system and the current auction information are updated.In one preferred embodiment, the auction server: adds thenewly-submitted bids that were entered in step 150 to the list ofprevious bids that remain in effect; replaces the previousprovisionally-winning bids with the provisionally-winning bids that werecalculated in the most recent execution of step 152; and replaces theprevious provisional revenues with the provisional revenues that werecalculated in the most recent execution of step 152. In a secondpreferred embodiment, the auction server additionally deletes some ofthe bids from the list of previous bids that remain in effect, in orderto reduce the size of the problem that the computer will face at thenext iteration of step 152. The process then loops to step 144.

If the auction should not continue, the process goes to step 160, inwhich a computer outputs a final message, including the allocation ofitems among bidders and the payments of the bidders. In one preferredembodiment, the auction server recalls its calculation of theprovisionally-winning bids at the most recent execution of step 152 andoutputs this in a final message as the determined allocation of itemsamong bidders and the payments of the bidders. The auction serveroutputs this final message through its network interface and transmitsit via the network. The bidder and auctioneer computers or terminalsthen receive the final message through their network interfaces anddisplay the information to bidders and the auctioneer through their userinterfaces. The process then ends.

Detail Elements Concerning Bidders Changing Flexible Bid Information

FIG. 14 a is a flow diagram illustrating an exemplary process by which abidder may enter flexible bid information into a computer database orchange existing flexible bid information. Thus, FIG. 14 a illustrates,in greater detail, step 146 of FIG. 13. The flexible bid information ofFIG. 14 a concerns the bidder's valuations for various items in theauction.

The process starts with step 202, in which bidder i selects a subset S⊂Ωof the set of all items being auctioned. In one preferred embodiment,bidder i enters his (or her) selection of subset S through the userinterface of his bidder computer or terminal, which then (if necessary)outputs his selection through its network interface and transmits it viathe network. The proxy agent of bidder i (if located on anothercomputer) then receives the selection of subset S through its networkinterface for use in the next step. In step 204, the proxy agent ofbidder i recalls the current valuation, v_(i)(S) (if any), currentlyassociated with subset S. In one preferred embodiment, the proxy agentof bidder i queries its database to obtain the current valuationv_(i)(S), and then (if necessary) outputs the current valuation v_(i)(S)through its network interface and transmits it via the network. Thebidder computer or terminal of bidder i then receives the currentvaluation v_(i)(S) through its network interface (if the proxy agent islocated on a different computer) and displays it on its user interface.In step 206, bidder i inputs a new valuation to be associated withsubset S (or cancels input of a new valuation for subset S). As before,in one preferred embodiment, bidder i enters the new valuation throughthe user interface of his bidder computer or terminal, which then (ifnecessary) outputs the new valuation through its network interface andtransmits it via the network. The proxy agent of bidder i (if located onanother computer) then receives the new valuation through its networkinterface for use in the following steps. In step 208, a computerdetermines whether changes to the flexible bid information of bidder iare allowed. In one preferred embodiment, the proxy agent of bidder imerely refers to a variable located in the memory of the same computeron which the proxy agent of bidder i resides. If this variable equalsone, then changes to the flexible bid information of bidder i areallowed; and if this variable equals zero, then changes to the flexiblebid information of bidder i are not allowed. If changes to the flexiblebid information of bidder i are allowed, the process continues with step210, where the proxy agent of bidder i sets v_(i)(S) equal to the newvaluation that was inputted for subset S in step 206. If changes to theflexible bid information are not allowed, or following step 210, theprocess goes to step 212, in which it is determined whether bidder iwishes to continue changing his flexible bid information. In onepreferred embodiment, the bidder computer or terminal of bidder idisplays this as a question through its user interface, bidder iresponds to this question through its user interface, and bidder i'sresponse is transmitted to any other components of the system requiringhis response through the network. If bidder i wishes to continuechanging his flexible bid information, the process loops back to step202; otherwise, the process ends.

FIG. 14 b is a flow diagram illustrating another exemplary process bywhich a bidder may enter flexible bid information into a computerdatabase or change his existing flexible bid information. Thus, FIG. 14b illustrates, in greater detail, step 146 of FIG. 13. The flexible bidinformation of FIG. 14 b may concern the bidder's valuations for variousitems in the auction or may concern a budget limit or parameter.

The process starts with step 252, in which bidder i indicates whether hewishes to change his valuation of a subset, or whether he wishes tochange his budget limit or parameter. If bidder i wishes to change hisflexible bid information for a valuation of a subset, then the processgoes to step 254, in which bidder i selects a subset S⊂Ω of the set ofall items being auctioned. In one preferred embodiment, bidder i entershis selection of subset S through the user interface of his biddercomputer or terminal, which then (if necessary) outputs his selectionthrough its network interface and transmits it via the network. Theproxy agent of bidder i (if located on another computer) then receivesthe selection of subset S through its network interface for use in thenext step. In step 256, the proxy agent of bidder i recalls the currentvaluation, v_(i)(S) (if any), currently associated with subset S. In onepreferred embodiment, the proxy agent of bidder i queries its databaseto obtain the current valuation v_(i)(S), and then (if necessary)outputs the current valuation v_(i)(S) through its network interface andtransmits it via the network. The bidder computer or terminal of bidderi then receives the current valuation v_(i)(S) through its networkinterface (if the proxy agent is located on a different computer) anddisplays it on its user interface. In step 258, bidder i inputs a newvaluation to be associated with subset S (or cancels input of a newvaluation for subset S). As before, in one preferred embodiment, bidderi enters the new valuation through the user interface of his biddercomputer or terminal, which then (if necessary) outputs the newvaluation through its network interface and transmits it via thenetwork. The proxy agent of bidder i (if located on another computer)then receives the new valuation through its network interface for use inthe following steps. In step 260, a computer determines whether changesto the flexible bid information of bidder i are allowed. In onepreferred embodiment, the proxy agent of bidder i merely refers to avariable located in the memory of the same computer on which the proxyagent of bidder i resides. If this variable equals one, then changes tothe flexible bid information of bidder i are allowed; and if thisvariable equals zero, then changes to the flexible bid information ofbidder i are not allowed if changes to the flexible bid information ofbidder i are allowed, the process continues with step 262, where theproxy agent of bidder i sets v_(i)(S) equal to the new valuation thatwas inputted for subset S in step 258. If changes to the flexible bidinformation are not allowed, or following step 262, the process goes tostep 272.

If bidder i wishes to change his flexible bid information for a budgetlimit or parameter, then the process goes to step 264, in which theproxy agent of bidder i recalls the current budget limit or parameter.In one preferred embodiment, the proxy agent of bidder i queries itsdatabase to obtain the current budget limit or parameter, and then (ifnecessary) outputs the current budget limit or parameter through itsnetwork interface and transmits it via the network. The bidder computeror terminal of bidder i then receives the current budget limit orparameter through its network interface (if the proxy agent is locatedon a different computer) and displays it on its user interface. In step266, bidder i inputs a new budget limit or parameter (or cancels inputof a new budget limit or parameter). In one preferred embodiment, bidderi enters the new budget limit or parameter through the user interface ofhis bidder computer or terminal, which then (if necessary) outputs thenew budget limit or parameter through its network interface andtransmits it via the network. The proxy agent of bidder i (if located onanother computer) then receives the new budget limit or parameterthrough its network interface for use in the following steps. In step268, a computer determines whether changes to the flexible bidinformation of bidder i are allowed. In one preferred embodiment, theproxy agent of bidder i merely refers to a variable located in thememory of the same computer on which the proxy agent of bidder iresides. If this variable equals one, then changes to the flexible bidinformation of bidder i are allowed; and if this variable equals zero,then changes to the flexible bid information of bidder i are notallowed. If changes to the flexible bid information of bidder i areallowed, the process continues with step 270, where the proxy agent ofbidder i sets the budget limit or parameter equal to the new value thatwas inputted in step 266. If changes to the flexible bid information arenot allowed, or following step 270, the process goes to step 272.

In step 272, it is determined whether bidder i wishes to continuechanging his flexible bid information. In one preferred embodiment, thebidder computer or terminal of bidder i displays this as a questionthrough its user interface, bidder i responds to this question throughits user interface, and bidder i's response is transmitted to any othercomponents of the system requiring his response through the network. Ifbidder i wishes to continue changing his flexible bid information, theprocess loops back to step 252; otherwise, the process ends.

Detail Elements Concerning Bid Submission by Proxy Agents

FIG. 15 a is a flow diagram illustrating an exemplary process by which aproxy agent may submit new bids based on a bidder's flexible bidinformation and the current auction information. Thus, FIG. 15 aillustrates, in greater detail, step 148 of FIG. 13. The flexible bidinformation of FIG. 15 a concerns the bidder's valuations for variousitems in the auction.

The process starts with step 302, in which the proxy agent of bidder iselects an arbitrary subset R⊂Ω of the set of all items being auctioned.Subset R is treated as the candidate package on which bidder i is to bid(until a better subset is found). The process goes to step 304, in whichthe proxy agent of bidder i selects a subset S⊂Ω that has not yet beenconsidered. At step 306, the proxy agent recalls the minimum bids,B_(i)(R) and B_(i)(S), that bidder i is permitted to place on subsets Rand S, respectively. In one preferred embodiment, the proxy agent ofbidder i queries a database as to the values of B_(i)(R) and B_(i)(S).(If the proxy agent of bidder i and the database containing the valuesof B_(i)(R) and B_(i)(S) are located on different computers, then thiscommunication occurs through the network interfaces of the respectivecomputers and via the network.) In another preferred embodiment, theproxy agent of bidder i outputs the query through the network interfaceof the computer on which it is located and transmits the query via thenetwork. The auction server then receives the query through its networkinterface (if located on another computer). The auction server thendetermines the values of B_(i)(R) and B_(i)(S) by calculations on datain the state of the auction system. The auction server then outputs thevalues of B_(i)(R) and B_(i)(S) through its network interface andtransmits them via the network (if necessary). The proxy agent of bidderi then receives the values of B_(i)(R) and B_(i)(S) through the networkinterface of the computer on which it is located (if the proxy agent islocated on a different computer), making it available for later steps.One exemplary calculation for determining the values of B_(i)(R) andB_(i)(S) is for the auction server to take the previous high prices bidfor R and S and to multiply each by a positive constant. A secondexemplary calculation for determining the value of B_(i)(R) is for theauction server to solve the following problem: what is the minimum bid(R, P) that could be submitted by bidder i such that, ifprovisionally-winning bids were calculated (see step 152 or 164, above)with the extra bid (R, P) included, then (R, P) would be aprovisionally-winning bid? (An analogous calculation would thendetermine B_(i)(S).) The process then goes to step 308, in which acomputer determines whether v_(i)(S)−B_(i)(S)>v_(i)(R)−B_(i)(R). In onepreferred embodiment, the proxy agent of bidder i merely refers tovariables v_(i)(R) and v_(i)(S), located in the memory of the samecomputer on which the proxy agent of bidder i resides, and performs thisdetermination. If v_(i)(S)−B_(i)(S)>v_(i)(R)−B_(i)(R), then the processgoes to step 310, where a computer sets R=S (i.e., subset S replacessubset R as the candidate package on which the proxy agent of bidder iis to bid). If v_(i)(S)−B_(i)(S)≦v_(i)(R)−B_(i)(R), or after step 310,the process continues to step 312, in which a computer determineswhether all subsets S⊂Ω have been considered. If not all subsets S⊂Ωhave been considered, the process loops back to step 304.

If all subsets S⊂Ω have been considered, the process goes to step 314,in which a computer determines whether v_(i)(R)−B_(i)(R)>0, that is,whether bidder i would receive positive surplus from a winning bid of(R, B_(i)(R)). If v_(i)(R)−B_(i)(R) is determined not to be greater thanzero, the process jumps to step 320, in which the proxy agent does notplace any new bids on behalf of bidder i, and the process ends. Ifv_(i)(R)−B_(i)(R) is determined to be greater than zero, the processcontinues to step 316, in which the proxy agent of bidder i determineswhether bidder i currently has a provisionally-winning bid on somepackage A at price P_(i)(A). In one preferred embodiment, the proxyagent of bidder i merely refers to variables, representing the currentprovisionally-winning bids of bidder i, located in the memory of thesame computer on which the proxy agent of bidder i resides, and performsthis determination. If bidder i does not currently have aprovisionally-winning bid, the process skips to step 322. If bidder idoes currently have a provisionally-winning bid on some package A atprice P_(i)(A), the process goes to step 318, in which a computerdetermines whether v_(i)(R)−B_(i)(R)>v_(i)(A)−P_(i)(A), that is, whetherbidder i would receive greater positive surplus from a winning bid of(R, B_(i)(R)) than from a winning bid of (A, P_(i)(A)). Ifv_(i)(R)−B_(i)(R) is determined not to be greater thanv_(i)(A)−P_(i)(A), the process continues to step 320, in which the proxyagent does not place any new bids on behalf of bidder i, and the processends. If v_(i)(R)−B_(i)(R) is determined to be greater thanv_(i)(A)−P_(i)(A), the process continues to step 322.

At step 322, the proxy agent submits a new bid on behalf of bidder i forpackage R at price B_(i)(R). In one preferred embodiment, the proxyagent of bidder i outputs the bid (R, B_(i)(R)) through the networkinterface of the computer on which it is located and transmits thesubmitted bid via the network. The auction server then receives thesubmitted bid through its network interface (if located on anothercomputer), and utilizes the submitted bid in subsequent steps (forexample, step 150 or step 162). After step 322, the process ends.

In another embodiment of the present invention, FIG. 15 a may bemodified so that the proxy agent of bidder i submits two or more newbids, if bidder i would be indifferent among these bids. Step 308 wouldbe expanded so that a computer determines whetherv_(i)(S)−B_(i)(S)=v_(i)(R)−B_(i)(R). In that event, step 310 wouldmaintain both R and S as candidate packages on which the proxy agent ofbidder i is to bid, and step 322 would have the proxy agent submit bidsboth of (R, B_(i)(R)) and (S, B_(i)(S)).

In other embodiments of the present invention, FIG. 15 a is easilymodified so that the proxy agent of bidder i bids on behalf of bidder iin a reverse auction or procurement auction. In one such embodiment,Step 306 is modified so that the bids, B_(i)(R) and B_(i)(S), aremaximum bids that bidder i is permitted to place on subsets R and S,respectively. Step 308 is modified so that a computer determines whetherB_(i)(S)−v_(i)(S)>B_(i)(R)−v_(i)(R), since B_(i)(R) and B_(i)(S) nowrepresent payments that the bidder is willing to accept, while v_(i)(R)and v_(i)(S) now represent costs of the bidder. Step 314 is modified sothat a computer determines whether B_(i)(R)−v_(i)(R)>0, since this nowdetermines whether bidder i would receive positive surplus from awinning bid of (R, B_(i)(R)). Step 318 is modified so that a computerdetermines whether B_(i)(R)−v_(i)(R)>B_(i)(A)−v_(i)(A), since this nowdetermines whether bidder i would receive greater positive surplus froma winning bid of (R, B_(i)(R)) than from a winning bid of (A, P_(i)(A)).

FIG. 15 b is a flow diagram illustrating another exemplary process bywhich a proxy agent may submit new bids based on a bidder's flexible bidinformation and the current auction information. Thus, FIG. 15 billustrates, in greater detail, step 148 of FIG. 13. The flexible bidinformation of FIG. 15 b concerns the bidder's valuations for variousitems in the auction and a budget limit or parameter.

The process starts with step 352, in which the proxy agent of bidder iselects a subset R⊂Ω of the set of all items being auctioned such thatthe minimum bid, B_(i)(R), that bidder i is permitted to place on subsetR is less than or equal to the budget limit or parameter of bidder i.(The proxy agent of bidder i recalls the minimum bid for subset R in thesame way as described in step 306 above. If no subset R exists such thatthe minimum bid, B_(i)(R), is within bidder i's budget limit orparameter, then the process jumps all the way to step 372 and does notsubmit any new bid for bidder i.) Subset R is treated as the candidatepackage on which bidder i is to bid (until a better subset is found).The process goes to step 354, in which the proxy agent of bidder iselects a subset S⊂Ω that has not yet been considered. At step 356, theproxy agent recalls the minimum bids, B_(i)(R) and B_(i)(S), that bidderi is permitted to place on subsets R and S, respectively. In onepreferred embodiment, the proxy agent of bidder i queries a database asto the values of B_(i)(R) and B_(i)(S). (If the proxy agent of bidder iand the database containing the values of B_(i)(R) and B_(i)(S) arelocated on different computers, then this communication occurs throughthe network interfaces of the respective computers and via the network.)In another preferred embodiment, the proxy agent of bidder i outputs thequery through the network interface of the computer on which it islocated and transmits the query via the network. The auction server thenreceives the query through its network interface (if located on anothercomputer). The auction server then determines the values of B_(i)(R) andB_(i)(S) by calculations on data in the state of the auction system. Theauction server then outputs the values of B_(i)(R) and B_(i)(S) throughits network interface and transmits them via the network (if necessary).The proxy agent of bidder i then receives the values of B_(i)(R) andB_(i)(S) through the network interface of the computer on which it islocated (if the proxy agent is located on a different computer), makingit available for later steps. One exemplary calculation for determiningthe values of B_(i)(R) and B_(i)(S) is for the auction server to takethe previous high prices bid for R and S and to multiply each by apositive constant. A second exemplary calculation for determining thevalue of B_(i)(R) is for the auction server to solve the followingproblem: what is the minimum bid (R, P) that could be submitted bybidder i such that, if provisionally-winning bids were calculated (seestep 152 or 164, above) with the extra bid (R, P) included, then (R, P)would be a provisionally-winning bid? (An analogous calculation wouldthen determine B_(i)(S).)

The process then goes to step 358, in which a computer determineswhether B_(i)(S) is less than or equal to the budget limit or parameterof bidder i. If B_(i)(S) is greater than bidder i's budget limit orparameter, then the process skips to step 364. If B_(i)(S) is less thanor equal to bidder i's budget limit or parameter, then the processcontinues to step 360, where a computer determines whetherv_(i)(S)−B_(i)(S)>v_(i)(R)−B_(i)(R). In one preferred embodiment, theproxy agent of bidder i merely refers to variables v_(i)(R) andv_(i)(S), located in the memory of the same computer on which the proxyagent of bidder i resides, and performs this determination. If v_(i)(S)−B_(i)(S)≦v_(i)(R)−B_(i)(R), then the process skips to step 364. Ifv_(i)(S)−B_(i)(S)>v_(i)(R)−B_(i)(R), then the process continues withstep 362, where a computer sets R=S (i.e., subset S replaces subset R asthe candidate package on which the proxy agent of bidder i is to bid),and then proceeds to step 364. At step 364, a computer determineswhether all subsets S⊂Ω have been considered. If not all subsets S⊂Ωhave been considered, the process loops back to step 354.

If all subsets S⊂Ω have been considered, the process goes to step 366,in which a computer determines whether v_(i)(R)−B_(i)(R)>0, that is,whether bidder i would receive positive surplus from a winning bid of(R, B_(i)(R)). If v_(i)(R)−B_(i)(R) is determined not to be greater thanzero, the process jumps to step 372, in which the proxy agent does notplace any new bids on behalf of bidder i, and the process ends. Ifv_(i)(R)−B_(i)(R) is determined to be greater than zero, the processcontinues to step 368, in which the proxy agent of bidder i determineswhether bidder i currently has a provisionally-winning bid on somepackage A at price P_(i)(A). In one preferred embodiment, the proxyagent of bidder i merely refers to variables, representing the currentprovisionally-winning bids of bidder i, located in the memory of thesame computer on which the proxy agent of bidder i resides, and performsthis determination. If bidder i does not currently have aprovisionally-winning bid, the process skips to step 374. If bidder idoes currently have a provisionally-winning bid on some package A atprice P_(i)(A), the process goes to step 370, in which a computerdetermines whether v_(i)(R)−B_(i)(R)>v_(i)(A)−P_(i)(A), that is, whetherbidder i would receive greater positive surplus from a winning bid of(R, B_(i)(R)) than from a winning bid of (A, P_(i)(A)). Ifv_(i)(R)−B_(i)(R) is determined not to be greater thanv_(i)(A)−P_(i)(A), the process continues to step 372, in which the proxyagent does not place any new bids on behalf of bidder i, and the processends. If v_(i)(R)−B_(i)(R) is determined to be greater thanv_(i)(A)−P_(i)(A), the process continues to step 374.

At step 374, the proxy agent submits a new bid on behalf of bidder i forpackage R at price B_(i)(R). In one preferred embodiment, the proxyagent of bidder i outputs the bid (R, B_(i)(R)) through the networkinterface of the computer on which it is located and transmits thesubmitted bid via the network. The auction server then receives thesubmitted bid through its network interface (if located on anothercomputer), and utilizes the submitted bid in subsequent steps (forexample, step 150 or step 162). After step 374, the process ends.

In another embodiment of the present invention, FIG. 15 b may bemodified so that the proxy agent of bidder i submits two or more newbids, if bidder i would be indifferent among these bids. Step 360 wouldbe expanded so that a computer determines whetherv_(i)(S)−B_(i)(S)=v_(i)(R)−B_(i)(R). In that event, step 362 wouldmaintain both R and S as candidate packages on which the proxy agent ofbidder i is to bid, and step 374 would have the proxy agent submit bidsboth of (R, B_(i)(R)) and (S, B_(i)(S)).

Core Outcomes and Bidder-Optimal Core Outcomes

It can be shown that outcomes of particular proxy auctions are elementsof the “core”, relative to the submitted bids, and Nash equilibriumoutcomes of particular proxy auction games are “bidder-optimal coreoutcomes”. We begin by defining these terms.

We define the coalitional game (L,w) that is associated with the tradingmodel. The set of players is L={0, 1, . . . , n}, with player 0 beingthe seller and players 1, . . . , n being the bidders.

The set of feasible allocations is X, for example,

$X = {\left\{ {\left( Q^{i} \right)_{i = 1}^{n}:{Q^{i} \geq {0\mspace{14mu}\text{and}\mspace{14mu}{\sum\limits_{i = 1}^{n}\; Q^{i}}} \leq \overset{\_}{Q}}} \right\}.}$The coalitional value function is defined for coalitions S⊂L as follows:

${w(S)} = \left\{ \begin{matrix}{{\max\limits_{Q \in X}{\sum\limits_{i \in S}{v^{i}\left( Q^{i} \right)}}},} & {{{{if}\mspace{14mu} 0} \in S},} \\{0,} & {{{if}\mspace{14mu} 0} \notin {S.}}\end{matrix} \right.$In this notation, if bidder i submitted the bid (Q^(i), P^(i)) in theproxy auction, then v^(i)(Q^(i))=P^(i). The value of a coalition is themaximum total value that the players can create by trading amongthemselves. If the seller is not included in the coalition, that valueis zero.

The core of a game with player set L and coalitional value function w(□)is defined as follows:

${{Core}\left( {L,w} \right)} = {\left\{ {{{\pi:{w(L)}} = {\sum\limits_{i \in L}^{\;}\;\pi^{i}}},{{{w(S)} \leq {\sum\limits_{i \in S}^{\;}\;{\pi^{i}\mspace{11mu}{for}\mspace{14mu}{all}\mspace{14mu} S}}} \Subset L}} \right\}.}$Thus, the core is the set of profit allocations that are feasible forthe coalition of the whole and unblocked by any coalition.

A payoff vector in the core is bidder optimal if there is no other coreallocation that all bidders prefer. More precisely, let πε Core (L, w).We say that π is bidder optimal in the core if there is no {circumflexover (π)}ε Core(L, w) with {circumflex over (π)}≠π and {circumflex over(π)}^(i)≧π^(i) for every bidder i=1, . . . , m.

The above definition of the core assumes “transferable utility”, i.e.,bidders have quasilinear utility. In the event that utility isnon-transferable, we should instead use the non-transferable-utility(NTU) core. An allocation Q is in the NTU core if: (1) it is feasible,(2) it is individually rational for each bidder and for the seller, and(3) there exists no coalition S and allocation {circumflex over (Q)}feasible for coalition S such that {circumflex over (Q)} is strictlypreferred to Q for all players i in coalition S. It can also be shownthat, in situations where utility is non-transferable, outcomes ofparticular proxy auctions are elements of the NTU core, relative to thesubmitted bids.

FIG. 16 is a flow diagram of an exemplary subprocess of step 134 of FIG.5, for an auction with m types of items. The process of FIG. 16 beginswith step 134 b-1, in which a bidder i who has not yet been consideredis selected. In step 134 b-2, a computer recalls the entire set Σ^(i) ofbids that were received from bidder i in the sealed bid phase of theauction and entered in step 132 of FIG. 5. If necessary, each bid(S^(i), P^(i)) in Σ^(i) is converted into the quantity vector notation,(Q^(i), P^(i)), where Q^(i)≡(Q₁ ^(i), . . . , Q_(m) ^(i)) and Q_(k) ^(i)denotes the quantity of items of type k in the set S^(i) of items.Meanwhile, P^(i) identifies a price for the entire package identified byQ^(i). The process continues to step 134 b-3, where the entirecollection of bidder i's quantity vectors and associated price vectors,{(Q^(i,s), P^(s))}, from the earlier phase of the auction are recalledand converted into package bids. Recall that (Q^(i,s), P^(s)) isconverted into a package bid (Q^(i,s), {circumflex over (P)}^(i,s)) bycalculating:

${\hat{P}}^{i,s} = {\sum\limits_{k = 1}^{m}\;{P_{k}^{s}{Q_{k}^{i,s}.}}}$The process then continues to step 134 b-4, where it is determinedwhether all bidders have been considered. If not, the process loops backto step 134 b-1.

If all bidders have been considered, the process goes to step 134 b-5,where a computer selects a bidder-optimal core outcome relative toreceived bids. In one preferred embodiment, a computer there selects abidder-optimal core outcome, relative to the received bids in thedynamic auction phase and the later phase of the auction. In a secondpreferred embodiment, a computer there selects a bidder-optimal coreoutcome, relative to the received bids in the later phase of the auctiononly. In each of these embodiments, there may be multiple bidder-optimalcore outcomes—in that event, a computer applies a tie-breaking rule fordetermining which bidder-optimal core outcome to select.

After selecting a bidder-optimal core outcome relative to received bids,the process continues to step 134 b-6, where the allocation of items andpayments of bidders implied by the selected bidder-optimal core outcomeis determined. After the allocation and payments have been determined,the process goes to step 136 of FIG. 5, where a computer outputs a finalmessage, including the allocation of items and payments of bidders.

Auction-Like Optimization Problems and Machine-Generated Bids

In the course of this application, a method and apparatus for a hybridauction including an earlier, dynamic auction phase, and a later,package auction phase, has been described. The method and apparatus thathave been described allow users to participate in various auctions witha level of attention that varies from continuous, down to the input ofinformation into a proxy agent on a single occasion. It should also beapparent that the required level of attention by the “auctioneer” mayvary from continuous to essentially zero—aside from setting the rulesfor initiating the auction. Thus for all intents and purposes, once thebasic auction description is selected and the users input desiredinformation, the auction implemented by the invention can be essentiallyautomatic, i.e., devoid of human interaction.

Because in the past auctions have generally been considered to beprocesses engaged in by persons, the feature of an automatic auction maybe, by itself, considered relatively new. There are, however, many otherautomatic systems which interact in a way which is entirely analogous toan auction and to which the present invention could be applied. Hence,the present invention can be applied to improve the efficiency ofcomputers which are used to operate the automatic systems, byeconomizing on the collection of informational inputs needed for thesystem and to speed the computational of optimal resource assignments.At the same time, many optimization problems encountered in the field ofoperations research have similar mathematical structures to the problemof determining the winners of an auction with package bidding. Hence,the present invention can be applied to improve the efficiency ofcomputer systems which are used to solve the similar optimizationproblems, by enabling the computations to be implemented on a systemwith parallel processing or generally by speeding the computation ofsolutions.

For example, the air conditioning plant in an office building canallocate cool air among individual offices in the building via a dynamicauction. Periodically, the central computer of the air conditioningsystem serves the role of the “auction computer” in an auction, whilecomputers interfaced with the thermostats in each suite of offices servethe role of “bidder computers.” Each bidder computer is programmed tosend back bids consisting of a desired quantity of cooled air based on:the current temperature reading of the thermostat, the desiredtemperature in the office, and the use (if any) to which the office iscurrently being put. In addition, it is desirable for the auction-likeautomatic system to allow package bidding, in the same way that it isdesirable for a conventional auction system for geographically-definedspectrum licenses to allow package bidding. (Cooling an individualoffice requires less cooled air if the adjacent offices are also beingcooled, just as the value of a New York-region spectrum license may beenhanced by owning a Washington-region spectrum license or aBoston-region spectrum license.) Based on the parameters to which it hasbeen programmed, the central computer of the air conditioning systemthen provides the results of the auction in its allocation of cooled airamong the bidding offices.

In another context, a communications, transportation or energytransmission network faces the technical problem of how to allocate itsscarce network resources. The optimization problem in allocating itsnetwork resources (e.g., bandwidth, switches, etc.) has a very similarstructure to the auction problem. Moreover, package bidding is againwell suited to the problem, since a network provider attempting toconnect point A to point B needs to utilize various networks links andswitches in combination. Hence, the present invention can be usefullyapplied to improving the solution to this technical problem.

In another context, computational resources on a distributed computersystem can be allocated via a dynamic auction. Whenever a new jobrequiring a given quantity of CPU time enters the system, an auction isconducted. Each member of the distributed computer system indicates thequantity of CPU time which it can make available at a given prioritylevel or a given price. In this case, the “auctioneer computer” selectsand allocates the resources to be applied to the new job in accordancewith some programmed schedule and hence in this fashion provides theresults of the auction.

The several examples described herein are exemplary of the invention,whose scope is not limited thereby but rather is indicated in theattached claims.

1. A method implemented in a system, said system comprising a firstcomputer and at least one other computer which is located remotely fromthe first computer and interconnected by a communication system, saidmethod for conducting an auction of a plurality of items wherein atleast the first computer receives bids and determines an allocation ofat least one of the items, the auction including a dynamic auction phasefollowed by a later phase, the later phase comprising a package auction,the method comprising: a) implementing the dynamic auction phase on saidfirst computer, said dynamic auction phase comprising: a1) receivingbids at the first computer from at least one bidder using the at leastone other computer, said bids including at least an indicator of atleast one of the items; a2) determining whether the dynamic auctionphase of the auction should continue, based on received bids; a3)outputting auction information; and a4) repeating a1)-a3) if the dynamicauction phase of the auction is determined to continue; b) changing fromthe dynamic auction phase to the later phase, following a determinationnot to continue the dynamic auction phase; and c) implementing the laterphase of the auction on said first computer, the later phase comprisinga package auction, said later phase comprising: c1) receiving bids atthe first computer from at least one bidder using the at least one othercomputer, said bids including at least an indicator of a package ofitems and an associated price for the package; and c2) determining anallocation of at least one of the items to one of the bidders based onreceived bids.
 2. A method as recited in claim 1 wherein each bidreceived in step a1) is a package bid including at least an indicator ofa package of items and an associated price for the package.
 3. A methodas recited in claim 2 wherein bids are constrained by an activity rulein the dynamic auction phase.
 4. A method as recited in claim 3 whereinbids are constrained by a revealed-preference activity rule in thedynamic auction phase.
 5. A method as recited in claim 1 wherein saidstep a1) includes transmitting a price vector to bidders prior toreceiving said bids.
 6. A method as recited in claim 5 wherein bids areconstrained by an activity rule in the dynamic auction phase.
 7. Amethod as recited in claim 6 wherein bids are constrained by arevealed-preference activity rule in the dynamic auction phase.
 8. Amethod as recited in claim 2 wherein the determining in the dynamicauction phase is based on solving a winner determination problem.
 9. Amethod as recited in claim 5 wherein the determining in the dynamicauction phase is based on comparing a sum of quantity vectors with anavailable quantity.
 10. A method as recited in claim 2 wherein thereceiving in the dynamic auction phase includes the receiving ofintra-round bids.
 11. A method as recited in claim 5 wherein thereceiving in the dynamic auction phase includes the receiving ofintra-round bids.
 12. A method as recited in claim 1 wherein the laterphase comprises a sealed bid package auction.
 13. A method as recited inclaim 12 wherein the determining in the later phase further includesdetermining a payment for each winning bidder.
 14. A method as recitedin claim 13 wherein the determined allocation of items and payments is acore outcome relative to the received bids in the later phase.
 15. Amethod as recited in claim 13 wherein the determined allocation of itemsand payments is a core outcome relative to the received bids in thedynamic auction phase and the later phase.
 16. A method as recited inclaim 13 wherein the determined allocation of items and payments is abidder-optimal core outcome relative to the received bids in the laterphase.
 17. A method as recited in claim 13 wherein the determinedallocation of items and payments is a bidder-optimal core outcomerelative to the received bids in the dynamic auction phase and the laterphase.
 18. A method as recited in claim 12 wherein bids in the laterphase are constrained by an activity rule.
 19. A method as recited inclaim 12 wherein bids in the later phase are constrained by a relaxedrevealed-preference activity rule.
 20. A method as recited in claim 1wherein the later phase comprises a dynamic package auction.
 21. Amethod as recited in claim 20 wherein the determining in the later phasefurther includes determining a payment for each winning bidder.
 22. Amethod as recited in claim 21 wherein the determined allocation of itemsand payments is a core outcome relative to the received bids in thelater phase.
 23. A method as recited in claim 21 wherein the determinedallocation of items and payments is a core outcome relative to thereceived bids in the dynamic auction phase and the later phase.
 24. Amethod as recited in claim 21 wherein the determined allocation of itemsand payments is a bidder-optimal core outcome relative to the receivedbids in the later phase.
 25. A method as recited in claim 21 wherein thedetermined allocation of items and payments is a bidder-optimal coreoutcome relative to the received bids in the dynamic auction phase andthe later phase.
 26. A method as recited in claim 20 wherein bids in thelater phase are constrained by an activity rule.
 27. A method as recitedin claim 20 wherein bids in the later phase are constrained by a relaxedrevealed-preference activity rule.
 28. A method as recited in claim 12wherein the later phase comprises a proxy auction.
 29. A method asrecited in claim 28 wherein bids in the later phase are constrained byan activity rule.
 30. A method as recited in claim 28 wherein bids inthe later phase are constrained by a relaxed revealed-preferenceactivity rule.
 31. A method as recited in claim 20 wherein the laterphase comprises a proxy auction.
 32. A method as recited in claim 31wherein bids in the later phase are constrained by an activity rule. 33.A method as recited in claim 31 wherein bids in the later phase areconstrained by a relaxed revealed-preference activity rule.
 34. A methodas recited in claim 12 wherein the determining in the later phase isbased on solving a winner determination problem.
 35. A method as recitedin claim 20 wherein the determining in the later phase is based onsolving a winner determination problem.
 36. A method as recited in claim28 wherein the determining in the later phase is based on solving awinner determination problem.
 37. A method as recited in claim 31wherein the determining in the later phase is based on solving a winnerdetermination problem.
 38. A computer implemented system for conductingan auction of a plurality of items wherein at least one computerreceives bids and determines an allocation of at least one of the items,the auction including a dynamic auction phase followed by a later phase,the later phase comprising a package auction, the system comprising: a)means for implementing the dynamic auction phase on a computer, saidmeans for implementing the dynamic auction phase comprising: a1 ) meansfor receiving bids from at least one bidder, said bids including atleast an indicator of at least one of the items; a2) means fordetermining whether the dynamic auction phase of the auction shouldcontinue, based on received bids; a3) means for outputting auctioninformation; and a4) means for repeating a1)-a3) if the dynamic auctionphase of the auction is determined to continue; b) means for changingfrom the dynamic auction phase to the later phase, following adetermination not to continue the dynamic auction phase; and c) meansfor implementing the later phase of the auction on a computer, the laterphase comprising a package auction, said means for implementing saidlater phase comprising: c1) means for receiving bids from at least onebidder, said bids including at least an indicator of a package of itemsand an associated price for the package; and c2) means for determiningan allocation of at least one of the items to one of the bidders basedon received bids.
 39. A system as recited in claim 38 wherein the meansfor receiving bids of a1) receives a package bid including at least anindicator of a package of items and an associated price for the package.40. A system as recited in claim 39 wherein the means for receiving bidsincludes means to constrain said bids by an activity rule.
 41. A systemas recited in claim 40 wherein the means for receiving bids includesmeans to constrain bids by a revealed-preference activity rule.
 42. Asystem as recited in claim 38 which further includes means fortransmitting a price vector to bidders and means for enabling the meansfor receiving bids to receive said bids only after said price vector hasbeen transmitted.
 43. A system as recited in claim 42 which includesmeans to constrain bids by an activity rule in the dynamic auctionphase.
 44. A system as recited in claim 43 wherein the means toconstrain bids constrains the bids by a revealed-preference activityrule.
 45. A system as recited in claim 39 wherein the means fordetermining of a2) solves a winner determination problem.
 46. A systemas recited in claim 42 wherein the means for determining of a2) comparesa sum of quantity vectors with an available quantity.
 47. A system asrecited in claim 39 wherein the means for receiving bids of a1) receivesat least intra-round bids.
 48. A system as recited in claim 42 whereinthe means for receiving bids of a1) receives at least intra-round bids.49. A system as recited in claim 38 wherein means for implementing thelater phase comprises means for implementing a sealed bid packageauction.
 50. A system as recited in claim 49 wherein the means fordetermining of c2) further includes means determining a payment for eachwinning bidder.
 51. A system as recited in claim 50 wherein the meansfor determining produces a core outcome relative to the received bids ofc1).
 52. A system as recited in claim 50 wherein the means fordetermining produces a core outcome relative to the received bids of a1)and c1).
 53. A system as recited in claim 50 wherein the means fordetermining produces a bidder-optimal core outcome relative to thereceived bids of c1).
 54. A system as recited in claim 50 wherein themeans for determining produces a bidder-optimal core outcome relative tothe received bids of a1) and c1).
 55. A system as recited in claim 49which includes means to constrain bids received by the means forreceiving of c1) by an activity rule.
 56. A system as recited in claim49 which includes means to constrain bids received by the means forreceiving of c1) by a relaxed revealed-preference activity rule.
 57. Asystem as recited in claim 38 wherein the means for implementing thelater phase implements a dynamic package auction.
 58. A system asrecited in claim 57 wherein the means for determining in the later phasefurther includes means for determining a payment for each winningbidder.
 59. A system as recited in claim 58 wherein the means fordetermining an allocation of items and payments determines a coreoutcome relative to the received bids in the later phase.
 60. A systemas recited in claim 58 wherein the means for determining an allocationof items and payments determines a core outcome relative to the receivedbids in the dynamic auction phase and the later phase.
 61. A system asrecited in claim 58 wherein the means for determining an allocation ofitems and payments determines a bidder-optimal core outcome relative tothe received bids in the later phase.
 62. A system as recited in claim58 wherein the means for determining an allocation of items and paymentsdetermines a bidder-optimal core outcome relative to the received bidsin the dynamic auction phase and the later phase.
 63. A system asrecited in claim 57 which further includes means for constraining bidsin the later phase by an activity rule.
 64. A system as recited in claim57 which further includes means for constraining bids in the later phaseby a relaxed revealed-preference activity rule.
 65. A system as recitedin claim 49 wherein the means for implementing the later phaseimplements a proxy auction.
 66. A system as recited in claim 65 whichfurther includes means for constraining bids in the later phase by anactivity rule.
 67. A system as recited in claim 65 which furtherincludes means for constraining bids in the later phase by a relaxedrevealed-preference activity rule.
 68. A system as recited in claim 57wherein the dynamic package auction implemented by the means forimplementing the later phase comprises a proxy auction.
 69. A system asrecited in claim 68 which further includes means for constraining bidsin the later phase by an activity rule.
 70. A system as recited in claim68 which further includes means for constraining bids in the later phaseby a relaxed revealed-preference activity rule.
 71. A system as recitedin claim 49 wherein the means for determining of c2) solves a winnerdetermination problem.
 72. A system as recited in claim 57 wherein themeans for determining of c2) solves a winner determination problem. 73.A system as recited in claim 65 wherein the means for determining of c2)solves a winner determination problem.
 74. A system as recited in claim68 wherein the means for determining of c2) solves a winnerdetermination problem.
 75. A computer system for determining anallocation of items and payments among a plurality of bidders whereinbids are received at the system and the allocation of the items and thepayments are determined by the system based on the received bids,comprising: means for receiving bids, including package bids for atleast two of the items, and means for processing the received bids todetermine an outcome including an allocation of the items among thebidders and payments associated with the bidders, wherein the determinedoutcome is a core outcome with respect to the received bids, said coreoutcome having an implied profit allocation that is feasible for thecoalition of the whole and unblocked by any coalition.
 76. A system asrecited in claim 75 wherein the determined outcome is a bidder-optimalcore outcome with respect to the received bids.
 77. A method fordetermining an allocation of items and payments among a plurality ofbidders, said method implemented in a system comprising a first computerand at least one other computer which is located remotely from the firstcomputer and interconnected by a communication system, wherein bids arereceived using the at least one other computer and the allocation of theitems and the payments are determined by the first computer based on thereceived bids, comprising: receiving bids, including package bids for atleast two of the items, using the at least one other computer,communicating the received bids to the first computer, and processingthe received bids using the first computer to determine an outcomeincluding an allocation of the items among the bidders and paymentsassociated with the bidders, wherein the determined outcome is a coreoutcome with respect to the received bids, said core outcome having animplied profit allocation that is feasible for the coalition of thewhole and unblocked by any coalition.
 78. A method as recited in claim77 wherein the determined outcome is a bidder-optimal core outcome withrespect to the received bids.
 79. A computer readable medium storing asequence of instructions which, when executed by a computer systemimplements a program for determining an allocation of items and paymentsamong a plurality of bidders wherein bids are received at the computersystem and the allocation of the items and the payments are determinedby the system based on the received bids, comprising: receiving bids,including package bids for at least two of the items, and processing thereceived bids to determine a core outcome with respect to the receivedbids, said core outcome including an allocation of the items among thebidders and payments associated with the bidders, whose implied profitoutcome is feasible for the coalition of the whole and unblocked by anycoalition.
 80. A computer readable medium as recited in claim 79 whereinthe determined outcome is a bidder-optimal core outcome with respect tothe received bids.